Smarandache 问题的研究

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This book can be ordered in a paper bound reprint from: The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212 USA Toll Free: 1-866-880-5373 E-mail: info@edupublisher.com Peer Reviewer: Prof. Wenpeng Zhang, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China. Prof. Xingsen Li, Ningbo Institute of Technology, Zhejiang University, Ningbo, P. R. China. Prof. Chunyan Yang and Weihua Li, Guangdong University of Technology, Institute of Extenics and Innovative Methods, Guangzhou, P. R. China. Prof. Qiaoxing Li, Lanzhou University, Lanzhou, P. R. China. Ph. D. Xiaomei Li, Dept. of Computer Science, Guangdong University of Technology, P. R. China. Copyright 2012 by The Educational Publisher, translators, editors and authors for their papers Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN: 9781599731902 Words: 175,000 Standard Address Number: 297-5092 Printed in the United States of America

8

š

8 c

š

Ăł . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

1 {0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 SmarandacheŸê†–SmarandacheŸê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1 'uSmarandacheŸê ˜‡Ă&#x;ÂŽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2 ˜aÂ?šSmarandacheŸêĂšEulerŸê Â?§I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3 ˜aÂ?šSmarandacheŸêĂšEulerŸê Â?§II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4 'u–SmarandacheŸê Ă&#x;ÂŽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 'uSmarandacheŸê†–SmarandacheŸê Â?§I . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Âľ9yG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 ˜ ƒ'ÂŻK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 'uSmarandacheŸê†–SmarandacheŸê Â?§II . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 'uSmarandacheŸêÂ†ÂƒĂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê . . . . . . . . . . . . . . . . 27 3.1 ˜a†–SmarandacheŸêƒ' ŸêÂ?§ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 'uSmarandachep‡Ÿê†–SmarandacheŸê Â?§ . . . . . . . . . . . . . . . . . . . . . 29 3.3 'ušÂ–SmarandacheŸê9Ă™ÊóŸê Â?§. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 'ušSmarandachep‡Ÿê†–SmarandacheŸê Â?§ . . . . . . . . . . . . . . . . . . 39 3.5 'uSmarandacheV ŒŸê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 ˜a2–SmarandacheŸê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 ˜‡Â?šSmarandacheŸê91 a–SmarandacheŸê Â?§ . . . . . . . . . . . . . 48 3.8 'uSmarandache LCMŸê†SmarandacheŸê ފ . . . . . . . . . . . . . . . . . . . . . . 51 3.9 'uSmarandache LCMŸê9Ă™ÊóŸê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 SmarandacheS ĂŻĂ„ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Smarandache LCM 'ÇS I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Smarandache LCM 'ÇS II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Smarandache 1 ÂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.1 SmarandacheĂŒÂ‚1 ÂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.2 Smarandache VĂŠÂĄ1 ÂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 Smarandache ĂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Smarandache 3n ĂŞiS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 I

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4.6 Smarandache kn ĂŞiS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.7 Smarandache ²Â?ĂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5 Ă™§êĂ˜ÂŻK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.1 škFibonacciê†LucasĂŞ Ă° ÂŞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 Dirichlet L-Ÿê†n Ăš . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 2Ă‚Dirichlet L-Ÿê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.1 'u½n5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.2 'u½n5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 ĂŤ Šz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

II

c

ó

c

ó

êØq¡ êØ, ´ïÄê 5Æ, AO´ïÄ ê5 êÆ©|. êØ/¤ Õá Æ

, XÙ¦êÆ©| uÐ, ïÄêØ { 3Øä uÐ, y

êØ®² \ êÆ Nõ©|, 3¥I, êØ ´uÐ @ êÆ©| . F1 <Ú¥I< é@Òk êØ £. êØ3êÆ¥ / ´ÕA , pdQ²`L/êÆ´ Æ

, êØ´êÆ¥

)0. Ïd, êÆ[ÑU rêØ¥ ] û ¦J / )þ ²¾0, ± y< /Á 0. êØfm© ÿ´^È ín {ïÄ ê 5 , =Ð êØ.

5úúuÐÑy)ÛêØ, êêØ, |ÜêØ . § Ï ïÄ { ØÓ

Ù¶, ѱÓ{ Ù Ä nØ â. Ö Ì éSmarandache¼ ê 9 Ù ' ¯ K C Ï ï Ä ( J ? 1 n n ã, Ì 0 Smarandache¼ ê 9 Ù § ' ¼ ê ( Ü ï Ä ? Ð, Ó 0

Smarandache ¼ê9 Ù§ '¼êéX #ïÄ(J,

0 A )Ûê

Ø¥ ¯K¿JÑ # ¯K. äN5`, ÖÌ XeSü: 1 ÙÌ {ü0 Smarandache¼ê9 Smarandache¼ê µÚ½Â; 1 ÙÌ 0 Smarandache¼ê Smarandache¼ê ïÄ(J9 # ¯K; 1n ÙÌ 0 Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê ïÄ( J; 1oÙÌ 0 SmarandacheS ïĤJ; 1ÊÙÌ 0 êإ٧S 9Dirichlet L-¼ê A ïĤJ. Ö´ÄuSmarandache¼ê9Ù ' X êدK # ïĤJ ®o. 3 ¤L§¥, É Ü©+ Ç õg , éd öL« ~a . Ó ö a Ü ó ÆÄ:ïÄÄ7é Ö ]Ï.

¤k Ö Ñ

z P ÚÆ

)!Ó1Ú?6L«©% a . du öY²k , Ö¥J Ñy Ø, H2 Ö ö1µ .

III

c

贸

IV

1Â˜Ă™ {0

1Â˜Ă™ {0 35Â?kÂŻK, vk)‰6Â˜Ă–ÂĽ, {7ÛêZĂŚĂ?ϐĂ˜;[F.Smarandache ÇJĂ‘ 108‡ÿ™)Ăť ĂŞĂ˜ÂŻK, ÚÚü ¯þêĂ˜;[ , . Ă–ĂŒÂ‡0

Š¼k'SmarandacheŸê†–SmarandacheŸê9Ă™Âƒ'ÂŻK ĂŻĂ„Ăš#?Ă?. ĂŠ?Âż ĂŞnĂ?Âś SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!, = S(n) = min{m : m ∈ N, n|m!}. lS(n) ½Ă‚<‚N´íĂ‘XJn = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r LÂŤn IOŠ)ÂŞ, @o S(n) = max {S(pÎąi i )}. 1≤i≤r

ddĂ˜JOÂŽĂ‘S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 1, S(12) = 4, S(13) = 13, S(14) = 7, S(15) = 5, S(16) = 6 ¡ ¡ ¡ .

ĂŠ ? Âż ĂŞnĂ? Âś –SmarandacheÂź ĂŞZ(n)½ Ă‚ Â?  êmÂŚ n

, Ă˜ m(m+1) 2

=

m(m + 1) Z(n) = min m : m ∈ N, n

. 2

lZ(n) ½ Ă‚ N ´ Ă­ Ă‘Z(n) c A ‡ Š Â?: Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 8, Z(10) = 4, Z(11) = 10, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 31, ¡ ¡ ¡ .

' uS(n)ĂšZ(n) ÂŽ â 5 Â&#x;, N Ăľ Æ Ăś Ă‘ ? 1 ĂŻ Ă„, Âź Ă˜ k (

J. Ă–ĂŒÂ‡ÂŒ7ÚßaŸê9Ă™Âƒ'Ÿê  #ĂŻĂ„?Ă?‰˜‡XĂš 8Bo (. äN/`, Ă–ĂŒÂ‡Â?)SmarandacheŸê†–SmarandacheŸê ĂŻĂ„(J! †SmarandacheŸê9–SmarandacheŸêƒ'ÂŻKĂŻĂ„(JĂšSmarandacheS ĂŻĂ„ (J , Â?ĂŞĂ˜OĂ?ĂśJøÍ ] , Â?Â?ĂŻĂ„SmarandacheÂŻK ÆÜJø Šz Â?B.

1

Smarandache ¼ê9Ù '¯KïÄ

2

1 Ă™ SmarandacheŸê†–SmarandacheŸê

1 Ă™ SmarandacheŸê†–SmarandacheŸê 2.1 'uSmarandacheŸê ˜‡Ă&#x;ÂŽ ĂŠ?Âż ĂŞn, Ă?Âś F. SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!. ~ XS(n) c A ‡ ŠS(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 10, S(11) = 11, S(12) = 4, ¡ ¡ ¡ . 'uS(n) { ߎâ5Â&#x;, žÍ Šz[1, 7], ĂšpĂ˜2­E.

'uS(n) Â? Â?5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âź Ă˜ k (J, ~XF. Luca Ç3Šz[2]ÂĽ?Ă˜ Ÿê A(x) =

1X S(n) x n6x

Ăže. OÂŻK, ‰Ñ A(x) ˜‡ r Ăž. O. ӞŒ„3Šz[3]ÂĽy² ? ĂŞ X

n>1

nι S(1) ¡ S(2) ¡ ¡ ¡ S(n)

´ýÊĂ‚Ăą , ÂżJĂ‘ eÂĄ Ă&#x;Ăż: Ă&#x; ÂŽ : ĂŠ?¿¢êx > 1, kĂŹCúª X LS(x) ≥ ln S(n) = ln x − ln ln x + O(1). n6x

'uĂšÂ˜Ă&#x;ÂŽ, –8q vk<ĂŻĂ„, – 3yk Šz¼„vw k'(Ă˜. F. Luca Ç@Â?kÂŒUy²ln x − ln ln x´LS(x) Ăž., ´ÊJy²LS(x) e.. Ăš

Â˜ÂŻK´k¿Â , – ÂŒÂąÂ‡NĂ‘S(n)Ÿê ÊêފŠĂ™5Â&#x;. Â?d, ĂŠĂšÂ˜ÂŻ K, |^Ă? 9)Ă›Â?{?1ĂŻĂ„, ÂŒÂą Ă‘ ,Ă˜Ă“ (Ă˜, =Xe½n:

½ n 2.1 ĂŠu?¿¢êx > 1, kĂŹCúª X LS(x) ≥ ln S(n) = ln x + O(1). n6x

w,d½nĂ˜=`²Ă&#x;ÂŽ 1 Â‡ĂŒÂ‘ln ln x´Ă˜ 3 , Â?Ă’´`,Šz[3]ÂĽ Ă&#x; ÿ´Â†Ă˜ , ӞÂ?‰Ñ LS(x) (LÂŤ/ÂŞ. ,XJ|^ƒêŠĂ™ÂĽ Â?( J, Â„ÂŒÂą Â?°( ĂŹCúª, =Ă’´

3

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ê?¿ êk > 1, k 1X LS(s) ≥ ln S(n) = ln x + C + O x n6x

1 , lnk x

Ù¼CÂ?ÂŒOÂŽ ~ĂŞ. Â? y²½n2.1, I‡eÂĄ A‡Ún. Ăš n 2.1 ĂŠ?ÂżÂƒĂŞp, kĂŹCúª X ln p p

p6x

X

= ln x + O(1),

ln p = x + O

p6x

x . ln x

y²:ĂŤ Šz[4 − 6].

eÂĄ|^dĂšn† ‰Ñ½n y². Ă„k‰ÑLS(x) Ăž. O. ¯¢Þ, dS(n) Ă? 5Â&#x; , ĂŠ?Âż ĂŞnkS(n) 6 n, ¤¹dEulerŒÚú

ª(Í Šz[6]¼½n3. 1), k 1X 1X LS(x) = ln S(n) 6 ln n x x n6x n6x Z Z x X 1 1 x ln(x + 1) = ln ndy 6 ln ydy + x x 1 x 1 n6x

= ln x − 1 +

1 ln(x + 1) + . x x

= LS(x) =

1X ln S(n) 6 ln x + O(1). x

(2-1)

n6x

Ù g, Ó | ^ Þ ¥ Ú n 5 OS(x) ≥

1 x

P

LnS(n) e .. ĂŠ ? Âż

n6x

ĂŞn > 1, w,n– k˜‡ƒĂ?fp, Ă˜Â” n = p ¡ n1 , u´dS(n) 5Â&#x; S(n) > p ln S(n) = ln S(p ¡ n1 ) > ln p. l dĂ™ Š Ă° ÂŞ(ĂŤ Šz[6]½n3.17), k LS(x) 1X = ln S(n) x n6x

4

1 Ă™ SmarandacheŸê†–SmarandacheŸê

1 X ln p x pn1 6x     X X X X X X 1 = ln p ¡ 1+ 1¡ ln p − ln p ¡ 1  x √ √ √ √ x n6 xp p6 n n6 x p6 x n6 x p6 x     X X X √ x x x 1 = ln p ¡ + O(1) + 1¡ +O − ln p ¡ x + O(1)  x √ p n n ln x √ √ p6 x n6 x p6 x      X X X 1 X √  √ ln p 1   = x¡ ln p + x ¡ ln p + O( x) +O + O(x) − x ¡  x p √ √ √ n √ p6 x n6 x n6 x p6 x √ √ √ √ √ √ x 1 = x ¡ ln x + O( x) + x ln x + O(x) − x x+O x ln x √ √ = ln x + ln x + O(1)

>

= ln x + O(1). = LS(x) > ln x + O(1).

(2-2)

dª(2-1)9(2-2)k LS(x) = ln x + O(1). u´ ¤½n2.1 y².

2.2 ˜aÂ?šSmarandacheŸêĂšEulerŸê Â?§I ĂŠu?Âż ĂŞn, Ă?Âś SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!, = Ă’ ´S(n) = min{m : m ∈ N, n|m!}.SmarandacheÂź ĂŞ ˆ ÂŤ 5 Â&#x; ´ ĂŞ Ă˜ 9 Ă™ A

^+Â?ÂĽÂ˜Â‡Â›ŠĂš<'5 ïđK(ĂŤ Šz[1, 7]). ĂŠu ĂŞn, φ(n)´' un EulerŸê, Ăšp φ(n)LÂŤĂ˜ÂŒun…†npƒ ĂŞ ‡ê(ĂŤ Šz[4]). ' uÂ?šEulerŸêĂšSmarandacheŸê Â?§ φ(n) = S(nk )

(2-3)

),ÊþÆÜÑ?1 ĂŻĂ„, ˜ Ă? (J[9−12] . ÂŒÂą|^Ă? Â?{)ĂťT Â?§3k = 7ž ÂŚ)ÂŻK, =‰ÑeÂĄ ½n. ½ n 2.2 k = 7ž, Â?§Ď†(n) = S(nk )=k)n = 1, 80. Â? y²½n2.2, I‡eÂĄ A‡Ún.

5

Smarandache ¼ê9Ù '¯KïÄ

Ú n 2.2 Euler¼ ê È 5 ¼ ê, = é u ? ¿ p êmÚn, K kφ(mn) = φ(m)φ(n). Ú n 2.3 n = pα1 1 pα2 2 · · · pαk k ´ ên IO©)ª, Kk φ(n) =

k Y

pαi i −1 (pi − 1).

i=1

Ú n 2.4 n > 2 , K7k2|φ(n). y ² : (1)e ênkÛ Ïf, Ø p, Kp > 2 2|p − 1.qdÚn2.3 (p − 1)|φ(n),¤±´ 2|φ(n).

(2)e ênvkÛ Ïf, n > 2 , 7kn = 2k ,k´ u1 ê.dEuler¼ ê 5 ´ φ(n) = 2k−1 (2 − 1) = 2k−1 , ¤± k2|φ(n). d(1)Ú(2) , Ún2.4¤ á.

Ú n 2.5 n = pα1 1 pα2 2 · · · pαk k ´ ên IO©)ª, K S(n) = max{S(pα1 1 ), S(pα2 2 ), · · · , S(pαk k )}. y ² : ë ©z[7]. Ú n 2.6 éu êpÚ êk, kS(pk ) 6 kp.AO/, k < p , kS(pk ) = kp. y ² : ë ©z[7]. Ú n 2.7 y = pα−2 (p − 1) − αp, p ê, K α > 3 , ¼êy´üN4O . y ² : Ï y ′ = (p − 1)pα−2 ln p − p, α > 3 , ´ y ′ > 0, (ؤá. e¡ ѽn2.2 y².rk = 7 \ §φ(n) = S(nk )¥,= φ(n) = S(n7 ).

(2-4)

w,n = 1´ª(2-4) ). ±eÌ ?Øn > 1 6. n > 1 n = pα1 1 pα2 2 · · · pαk k ´ ên IO©)ª,KdÚn2.5k 7αk 7α2 7r 1 S(n7 ) = max{S(p7α 1 ), S(p2 ), · · · , S(pk )} = S(p ).

(2-5)

dÚn2.2 , r

φ(n) = φ(p )φ

n pr

r−1

=p

(p − 1)φ

n pr

.

(2-6)

éáª(2-4))ª(2-6) , r−1

p

(p − 1)φ

n pr

6

= S(p7r ).

(2-7)

1 Ă™ SmarandacheŸê†–SmarandacheŸê

r = 1ž, ep = 2,dÂŞ(2-7)ÂŒ n

S(27 ) = 8 = φ

2

=n = 32,

,

φ(32) = 25 − 24 = 16 6= S(327 ), n = 32Ă˜´ª(2-4) ). ep = 3,K S(37 ) = 18 = 2φ چÚn2.4gĂą, ¤¹ª(2-4)Ăƒ).

n

3

,

Ă“nÂŒ , ep = 5, 7, 11ž, ÂŞ(2-4)Ăƒ). ep > 11,K

n S(p ) = 7p = (p − 1)φ , p 7

Â…2|p − 1, چÚn2.4gĂą, ¤¹ª(2-4)Ăƒ). r = 2ž, ep = 2, dÂŞ(2-7)ÂŒ

S(214 ) = 16 = 2φ =φ( n4 ) = 8, Kn = 64,

n

4

,

S(647 ) = S(242 ) = 46 6= 32 = 26 − 25 = φ(64), ¤¹n = 64Ă˜´ª(2-4) ). ep = 3, K S(314 ) = 30 = 6φ چÚn2.4 gĂą, ¤¹ª(2-4)Ăƒ).

n

9

,

Ă“nÂŒy,ep = 5, 7, 11, 13,ÂŞ(2-4)Ăƒ). ep > 17,K 14

S(p ) = 14p = p(p − 1)φ =k p−1 7= φ 2

n p2

n p2

,

چÚn2.4gĂą, ÂŞ(2-4)Ăƒ). r = 3ž, ep = 2, 21

2

S(2 ) = 24 = 2 φ

7

n

8

,

,

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

K n8 = 9, ¤¹n = 72, φ(72) = (23 − 22 )(32 − 3) = 24, S(727 ) = S(221 ¡ 314 ) = 30 6= 24 = φ(72), n = 72Ă˜´ª(2-4) ). ep = 3,K S(321 ) = 45 = 18φ چÚn2.4gĂą, ÂŞ(2-4)Ăƒ).

n

27

,

ep = 5, 7, 11, 13, 17, 19,Ă“nÂŒyÂŞ(2-4)Ăƒ). ep > 23,dĂšn2.6k, 21

2

S(p ) = 21p = p (p − 1)φ = 21 = p(p − 1)φ

n p3

n p3

,

,

چp > 23gĂą, ÂŞ(2-4)Ăƒ). r = 4ž, ep = 2, S(228 ) = 32 = 23 φ =φ

n 16

= 4, ¤¹n = 80,

n

16

,

φ(80) = (24 − 23 )(5 − 1) = 32, S(807 ) = S(228 ¡ 57 ) = 32 = φ(80),

n = 80´ª(2-4) ). ep = 3, n S(3 ) = 60 = 3 ¡ 2φ , 81 28

3

Ú´Ă˜ÂŒU , Ă?dÂŞ(2-4)Ăƒ). ep > 5,dĂšn2.6k,

28

3

28p > S(p ) = p (p − 1)φ = 2

28 > p (p − 1)φ

n p4

n p4

,

چp > 5gĂą, ÂŞ(2-4)Ăƒ). r = 5ž,ep = 2, S(235 ) = 42 = 24 φ

8

n

32

,

,

1 Ă™ SmarandacheŸê†–SmarandacheŸê

چÚn2.4gĂą, ÂŞ(2-4)Ăƒ). ep > 3, 35

4

35p > S(p ) = p (p − 1)φ = 3

35 > p (p − 1)φ

n p5

n p5

,

,

چp > 3gĂą, ÂŞ(2-4)Ăƒ). r > 6ž,ep = 2, S(27r ) = 2r−1 φ

n

2r

,

=‡2r−1 |S(p7r ), dSmarandacheŸê ½Ă‚ÚÚn2.5!2.6 ÂŒ , ĂšĂ˜ÂŒU¤å, ¤¹ ÂŞ(2-4)Ăƒ).

ep > 3,K 7r

r−1

7rp > S(p ) = p = r−2

7r > p

(p − 1)φ

(p − 1)φ

n pr

n pr

,

> pr−2 (p − 1),

dĂšn2.7ÂŒ ,ĂšĂ˜ÂŒU¤å, ¤¹ª(2-4)Ăƒ). nÞ¤ã: k = 7ž, Â?§Ď†(n) = S(nk )=k)n = 1, 80.Ă?d ¤ ½n2.2 y ².

2.3 ˜aÂ?šSmarandacheŸêĂšEulerŸê Â?§II ĂŠ u ĂŞn, SmarandacheÂź ĂŞ Â?S(n), φ(n)´EulerÂź ĂŞ, ! U Y ĂŁ Â? šSmarandacheŸêĂšEulerŸê Â?§ φ(n) = S(nk ) ). 3ĂžÂ˜!ÂĽ, Ž²Â‰Ă‘ Â?§(2-3)3k = 7ž Ăœ), !ò3dĂ„:މÑÂ? §(2-3)3k > 8ž ÂŚ)L§¹9Â?§) ‡ê¯K. =kXeA‡½n. ½ n 2.3 k = 8ž, Â?§(2-3)=k)n = 1, 125, 250, 289, 578. ½ n 2.4 k = 9ž, Â?§(2-3)=k)n = 1, 361, 722. ½ n 2.5 k > 10ž, Â?§(2-3)= 3k Â‡ ĂŞ). ½ n 2.6 en = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąs s ´ ĂŞn IOŠ)ÂŞ, Â… kÎą2 kÎąs 1 S(pkr ) = max{S(pkÎą 1 ), S(p2 ), ¡ ¡ ¡ , S(ps )},

9

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

K p > 2k + 1Â…k > 1, r > 2ž, Â?§(2-3)Ăƒ); 2k + 1Â?ÂƒĂŞÂž, Â?§(2-3) =k2‡ ), Â…n = 2p2 . eÂĄ5Â˜Â˜Â‰Ă‘¹Þ½n y². Ă„k½n2.3 y². rk = 8“\Â?§(2-3)ÂŒ , φ(n) = S(n8 ).

(2-8)

w,n = 1´ª(2-8) ).ÂąeĂŒÂ‡?Ă˜ n > 1ž Âœ6. n > 1Â…n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąs s ´ ĂŞn IOŠ)ÂŞ, KdĂšn2.6k 8Îąk 8Îą2 8r 1 S(n8 ) = max{S(p8Îą 1 ), S(p2 ), ¡ ¡ ¡ , S(pk )} = S(p )

(2-9)

dĂšn2.3ÂŒ , r

φ(n) = φ (p ) φ

n pr

r−1

=p

(p − 1)φ

n pr

.

(2-10)

ĂŠĂĄ(2-8)-(2-10)ÂŞÂŒ , r−1

p

(p − 1)φ

n pr

= S(p8r ).

(2-11)

p = 2, r = 1ž,d(2-11)ÂŒ n φ = S(28 ) = 10, 2 d(2-12)ÂŞÂŒĂ­ φ n2 > 1. Kn7šĂ›ÂƒĂ?fq,Â…   18, q=3      35, q=5 8 S q =  49, q=7      8q, q>7

(2-12)

(2-13)

¤¹(2-9)ÂŞĂš(2-12)ÂŞgĂą. p = 2, r = 1ž, ÂŞ(2-8)Ăƒ). φ(32) = 25 − 24 = 16 6= S(327 ), n = 32Ă˜´ª(2-8) ).

p = 2, r = 2ž, d(2-11)ÂŒ 2φ( n4 ) = S(216 ) = 18, =φ( n4 ) = 9, †Ún2.4gĂą, ¤¹ p = 2, r = 2ž, ÂŞ(2-8)Ăƒ). Ă“nÂŒy, p = 2, r = 3, 4, 5, 6, 7, 8ž, ÂŞ(2-8)Ăƒ). p = 2, r > 8ž, dĂšn2.6k n n 1 1 1 8r > S(28r ) = 2r−1φ r = 2r−2 φ r > 2r−2 = 2r , 2 2 2 2 4

=32r > 2r , gĂą. ¤¹ p = 2, r > 8ž, ÂŞ(2-8)Ăƒ). p = 3, r = 1ž, d(5)ÂŞÂŒ 22 φ n3 = S 38 = 18, =φ Ăą,¤¹ p = 3, r = 1ž, ÂŞ(2-8)Ăƒ).

10

n 3

= 9, †Ún2.4g

1 Ă™ SmarandacheŸê†–SmarandacheŸê

Ă“nÂŒy, p = 3, r = 2, 3, 4ž,ÂŞ(2-8)Ăƒ). p = 3, r > 5ž, dĂšn2.6k n n 1 2 8r > S(28r ) = 3r−1 φ r = 2 ¡ 3r−2 φ r > 2 ¡ 3r−2 > 2er−2 > 3 3 3 3 1 1 1 1 2 3 4 5 2 1 + (r − 2) + (r − 2) + (r − 2) + (r − 2) + (r − 2) > 8r, 2 6 24 120

gĂą.¤¹ p = 3, r > 5ž,ÂŞ(2-8)Ăƒ). p = 5, r = 1ž, d(2-11)ÂŞÂŒ 4φ n5 = S(58 ) = 35,gĂą, ÂŞ(2-8)Ăƒ). n n p = 5, r = 2ž,20φ 25 = S(516 ) = 70, =2φ( 25 ) = 7, gĂą, ÂŞ(2-8)Ăƒ). n n p = 5, r = 3ž,100φ( 125 ) = S(524 ) = 100, =φ 125 = 1, ¤¹n = 125, 250,

φ(125) = φ(53 ) = 100 = S(524 ) = S(1258 ),

φ(250) = φ(2 ¡ 53 ) = 100 = S(524 ) = S(2508 ), n = 125, 250´ª(2-8) ). p = 5, r = 4ž, Ă?Â?53 ¡ 4φ

n 625

p = 5, r > 4ž, ÂŞ(2-8)Â?Ăƒ).

p = 7, r = 1ž,d(2-11)ÂŞÂŒ 6φ

> S(532 ) = 130,¤¹ª(2-8)Ăƒ). Ă“nÂŒ n 7

= S(78 ) = 49,gĂą, ÂŞ(2-8)Ăƒ).

Ă“nÂŒ p = 7, r = 2ž, (2-8)ÂŞĂƒ). p = 7, r = 3ž,Ă?Â? n 72 ¡ 6φ > S(724 ) = 147, 243

¤¹(2-8)ÂŞĂƒ); p = 7, r > 3ž, ÂŞ(2-8)Ăƒ); p = 11, 13, r > 1ž, ÂŞ(2-8)Ăƒ). n p = 17, r = 1ž,d(2-11)ÂŞÂŒ 16φ 17 = S 178 = 17 ¡ 8,gĂą, ÂŞ(2-8)Ăƒ

).

p = 17, r = 2ž,k17¡16φ

n 172

= S 1716 = 17¡16,=φ

n 289

= 1, n = 289, 578,

φ(289) = φ 172 = 272 = S 1716 = S(2898 ), φ(578) = 272 = S 1716 = S 5788 ,

n = 289, 578´ª(2-8) ).

p = 17, r > 3ž, du17r−1 ¡ 16φ 17nr > S 178r ,¤¹ª(2-8)Ăƒ). n p = 19, r = 1ž,k18φ 19 = S 198 = 8 ¡ 19,gĂą,¤¹ª(2-8)Ăƒ).

Ă“n, p = 19, r = 2, 3ž,ÂŞ(2-8)Ăƒ); p = 19, r > 4ž, 19r > 19r−2 18φ( 19nr ) >

18 ¡ 19r−2 ,gĂą. ÂŞ(2-8)Ăƒ).

p > 19, r = 1ž,k(p − 1)φ( np ) = S(p8 ) = 8 ¡ p,Ă? Â?(p, p − 1) = 1, Â… e 11

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„ 8p k p−1 = 2k(k ∈ N + ), =p =

k(p−1) , 4

gĂą.¤¹ª(2-8)Ăƒ).

p > 19, r > 2ž,dĂšn2.6k r−2

8r > p

(p − 1)φ

n pr

> 22 ¡ 23r−2,

w,gĂą. ÂŞ(2-8)Ăƒ). nÞ¤ãÂŒ : k = 8ž,ÂŞ(2-8)=k)n = 1, 125, 250, 289, 578.Ăš Ă’y² ½ n2.3. Ă“n,^ƒq Â?{ÂŒÂąy²½n2.4Â?¤å. eÂĄy²½n2.5. ĂŠ?¿‰½ ĂŞk, n > 1, Â…n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąs s ´

ĂŞn IOŠ)ÂŞ, dS(n) ½Ă‚ÂŒ

kι2 kιs kr 1 S(nk ) = max{S(pkι 1 ), S(p2 ), ¡ ¡ ¡ , S(ps )} = S(p ).

qĂ? r

φ(n) = φ(p )φ

n pr

r−1

=p

(p − 1)φ

n pr

,

(2-14)

dÂ?§(2-3)ÂŒ r−1

p

n (p − 1)φ r p

= S pkr .

(2-15)

p 6 S(pkr ) 6 krp. epr−1 (p − 1)φ( pnr ) = krp,Kdφ(n) ½Ă‚95Â&#x;ÂŒ pr−1(p − 1)|krp,=pr−2 (p −

1)|kr.-y = pr−2 (p − 1) − kr, pÂ?‰½ ÂƒĂŞ,y´r Ÿê.K y ′ = (p − 1)pr−2 ln p − k.

p > 2k + 1Â…r > 2ž,ÂŒ y ′ > 0, džy´ßN4O ,Â…´ y > 0. ¤¹ª(2-15) p > 2k +1Â…r > 2ÂžĂƒ).== 3k Â‡ ĂŞnUÂŚ ÂŞ(2-13)¤ ĂĄ. Â

, ÂŒÂądĂšn2.2-2.6,(Ăœ½n2.4!½n2.5 y²Â?{Ă­Ă‘½n2.6Â?¤å.

2.4 'u–SmarandacheŸê Ă&#x;ÂŽ SmarandacheŸê´dF.Smarandache Ç3≪ Only Problems Not Solutions â‰ŤÂ˜

Ă–ÂĽĂš? , Ă™½Ă‚Â?S(n) = min{m : n|m!}. 5<‚Â?âSmarandacheŸê½Ă‚

n o m(m+1) –Smarandache ŸêZ(n) = min m : n| 2 .§Â‚kNĂľk 5Â&#x;, NĂľ<QĂŠ 12

1 Ă™ SmarandacheŸê†–SmarandacheŸê

d?1LĂŻĂ„. Erd¨ os[15] Q²Ă&#x;ÿÊuA ¤k nĂ‘kS(n) = P (n),Ù¼P (n)´Â?n  ÂŒÂƒĂ? fŸê. ĂŠuڇĂ&#x;Ăż8c¤Ÿ  Ă?(J´Aleksandar Ivic[25] , ÂŚy²

p log log log x N(x) = x exp − 2 log x log log x 1 + O , log log x

Ăš pN(x)L ÂŤ ¤ k Ă˜ ‡ Lx ĂŞ ÂĽ Ă˜ á v Â? §S(n) = P (n) ĂŞ ‡ ĂŞ.Mark FarrisĂšPatrick Mitchell[26] ĂŻĂ„ SmarandacheŸêS(n)3ÂƒĂŞÂ˜Ăž Ăže. O, Âż Xe(J: (p − 1)Îą 6 S(pÎą ) 6 (p − 1)(Îą + 1 + logÎąp ) + 1; Jozsef Sandor[27] ĂŻĂ„ Â?šSmarandacheŸêĂ˜ ÂŞ ÂŒ)5, y²

S(m1 + m2 + ¡ ¡ ¡ + mk ) < S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ) Ăš S(m1 + m2 + ¡ ¡ ¡ + mk ) > S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ) 3ĂƒÂĄĂľ). Lu Yaming[28] ?Â˜Ăšy² 3 Ă’¤åeÞãÂ?§Â? 3ĂƒÂĄĂľ|). 'uS(n)ĂšZ(n)„kNĂľ5Â&#x;ĂżĂ˜Â˜Ă™, AO´Z(n)duي ŠĂ™ Ă˜5K5 ÂŚ ĂŻĂ„ĂĽ5äkĂŠÂŒ (J.ĂœŠ+3[29]ÂĽJĂ‘e ÂŻK:´Ă„ þÂ?kk Â‡ P 1 ĂŞÂŚ n|d S(d) Â? ĂŞ; ŸêZ(n) ފ´þ ÂşMajumdar3[30]ÂĽJĂ‘ Âąeo

‡Ă&#x;ÂŽ:

(1)Z(n) = 2n − 1 Â…= n = 2k ,Ù¼kÂ?šK ĂŞ;

(2)Z(n) = n − 1 Â…= n = pk ,Ù¼pÂ?ÂŒu3 ÂƒĂŞ; (3)XJnĂ˜ULˆÂ?n = 2k /ÂŞ, KkZ(n) 6 n − 1;

(4)Ê?¿ nÑkZ(n) 6= Z(n + 1).

!òĂŒÂ‡ ã¹Þ¯K )‰, Ă˜ ĂŠuĂœŠ+¤JĂ‘ ÂŻK‰Ñ ˜½)‰, „ Majumdar¤JĂ‘ o‡Ă&#x;ÂŽ´ ( . Â? `²Ă&#x;ÂŽ (5, k‰ÑeÂĄA ‡Ún. Ăš n 2.8 eS(n) = P (n), n IOŠ)n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r P Îąr+1 (n), @o7kÎąr+1 = 1;ĂŠu1 6 k 6 r,7kÎąk 6 P (n) − 2.

y ² : eÎąr+1 > 1, KkP 2 (n)|n, n|S(n)! = P (n)!,KP 2 (n)|P (n)!.چP (n)Â?ÂƒĂŞg Ăą. kÎąr+1 = 1. n!ÂĽÂ?šÂƒĂ?fp ‡êÂ?Îą,=ávpÎą |n!, pÎą+1 ∤ n!,KkÎą =

13

P∞ h n i l=1 pl (y²Í

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

„[31]). dukpιk k |P (n)!,�dk ιk 6

∞ X P (n)

plk

l=1

6

∞ X P (n) l=1

plk

=

P (n) , pk − 1

Ă?d pk 6= 2ž´w,kÎąk 6 P (n) − 2; pk = 2ž, KktÂŚ 2t < P (n) < 2t+1 ,d žk

Îąk 6

t X P (n) l=1

2l

6

t X P (n) l=1

2l

= P (n) − P (n)

2 2t+1

< P (n) − 1,

Ă?Â?Îąk Â? ĂŞĂ?dkÎą 6 P (n) − 2.

Â? Â?\Â?B ĂŻĂ„Z(n), ½Ă‚˜‡# ŸêZ ∗ (n) = min{m : n|m(m + 1)}, w,

N´ nÂ?ÛêžZ(n) = Z ∗ (n), nÂ?óêžZ(n) = Z ∗ (2n), Ă?dÚ߇Ÿê´Â— ƒƒ' .e¥‰Ñ'uŸêZ ∗ (n) 5Â&#x;. Ăš n 2.9 n = pk ž, Z ∗ (n) = n−1; nšk߇¹ÞĂ˜Ă“ ƒĂ?fž, K7kZ ∗ (n) 6 n 2

− 1.

y ² :dZ ∗ (n) ½Ă‚´ Z ∗ (n) 6 n − 1, n = pk žkpk |Z ∗ (n)(Z ∗ (n) + 1), k pk |Z ∗ (n) ½ pk |Z ∗ (n) + 1, dÂžĂ‘ÂŒÂą Z ∗ (n) > pk − 1 = n − 1, džkZ ∗ (n) = n − 1.

nšk߇¹ÞĂ˜Ă“ ƒĂ?fž, džn7ÂŒŠ)Â?߇pƒ…Œu1 ĂŞ

ÂŚĂˆ, n = mk,Ù¼(m, k) = 1,džŒ kÂ?Ûê, K7½ 3Â…Â?˜ 3b, b′ ĂĄ uk  Âš K Â? { X ÂŚ mb ≥ −l mod kÂ…mb′ ≥ l mod k, Ă? d 7 kZ ∗ (n) 6

min{mb, mb′ − 1}. 5 Âż k|(mb + mb′ ),Ă? d kk|(b + b′ ), kb + b′ = k,d ukÂ? Ă› ĂŞ, 7kmin{b, b′ } 6

(k−1) ,Ă?d 2

Z ∗ (n) 6 m

k−1 n m n = − 6 − 1. 2 2 2 2

d¹Þ ĂšnÂŒÂą eÂĄA‡½n. ½ n 2.7 f (n) =

X 1 , S(d) d|n

@o3n 6 x ĂŞÂĽĂ˜ N(x)‡ ĂŞ , f (n)ĂžĂ˜ êŠ, Ù¼ p log log log x N(x) = x exp − 2 log x log log x 1 + O , log log x 14

1 Ă™ SmarandacheŸê†–SmarandacheŸê

=f (n)3A ¤k ĂŞĂžĂ‘Ă˜ êŠ. y ² :dAleksandar Ivic[25] (J, Â?Iy² S(n) = P (n)ž, f (n)Ă˜Â? ĂŞ. eS(n) = P (n),dĂšn2.8 , ÂŒ n IOŠ)Â?n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r P (n), Ù¼ιk 6

P (n) − 2,ònPÂ?n = mP (n),¤¹džk X 1 X 1 X 1 f (n) = = + S(d) S(d) S(dP (n)) d|n

d|m

d|m

X 1 X 1 = + S(d) P (n) d|m d|m Q r X 1 (Îąk + 1) = + k=1 . S(d) P (n) d|m

Â?I5Âż ĂŠud|m, S(d) < S(n) = P (n), Îąk + 1 6 P (n) − 1 < P (n), Ă?dĂŠu1˜ ĂœŠ, òĂ™Ă?Š

, Ă™Š1 ƒĂ?f uP (n),ĂŠu1 ĂœŠĂ™Šf ƒĂ?fÂ?

uP (n),Ă?dْ½Ă˜´ ĂŞ, Ă?dÚßĂœŠÂƒ\Ă˜ÂŒU¤Â? ĂŞ.½n2.7 y. ½ n 2.8 (i)Z(n) = 2n − 1 Â…= n = 2k , Ù¼kÂ?šK ĂŞ;

(ii) Z(n) = n − 1 Â…= n = pk , Ù¼pÂ?ÂŒu2 ÂƒĂŞ;

(iii)XJnĂ˜ULˆÂ?n = 2k /ÂŞ, KkZ(n) 6 n − 1.

y ² :(i) n = 2k ž(= Ă„kÂ? ĂŞ, k = 0´w, ), džkZ(n) = Z ∗ (2n), dĂš n2.9ÂŒ Z(n) = 2n − 1; eZ(n) = 2n − 1,7knÂ?óê, Ă„KZ(n) = Z ∗ (n) 6 n − 1, b n 6= 2k , 2n– „škÂ˜Â‡Ă›ÂƒĂ?f, dĂšn2.9ÂŒ Z(n) = Z ∗ (2n) 6 gĂą, n = 2k .

2n −1 2

= n−1,

(ii) n = pk , Â…pÂ?ÂŒu2 ÂƒĂŞÂž, džkZ(n) = Z ∗ (n), dĂšn2.9ÂŒ Z(n) = n − 1; eZ(n) = n − 1,w,nĂ˜ÂŒU´óê, b n 6= pk ,Kn– škĂźÂ‡Ă›ÂƒĂ?f, d Ăšn2.7ÂŒ Z(n) = Z ∗ (n) 6

n 2 − k

1,gĂą, kn = pk , Â…pÂ?ÂŒu2 ÂƒĂŞ.

(iii)enĂ˜ULˆÂ?n = 2 /ÂŞ, K7knÂ?Ûê, džkZ(n) = Z ∗ (n) 6 n − 1,

½KnÂ?óê…– škÂ˜Â‡Ă›ÂƒĂ?f, dĂšn2.9ÂŒ džk Z(n) = Z ∗ (2n) 6

2n − 1 = n − 1. 2

½ n 2.9 ĂŠ?Âż nĂ‘kZ(n) 6= Z(n + 1). y ² :b knÂŚ Z(n) = Z(n + 1) = m,K 3k, k ′ ÂŚ nk = (n + 1)k ′ =

m(m+1) ,d 2

u(n, n+1) = 1,Ă?dkn+1|k, n|k ′ , Ă?dkn+1 6 k, n 6 k ′ , ÂŒ m(m+1) > n(n+1), 2 ′

km > n + 1,d½n2.8ÂŒ K7kn = 2t , n + 1 = 2t , Â?U´n = 1, džw, kZ(1) 6= Z(2). 15

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

½ n 2.10 Ê?¿ x > 1, k 2 X J X 3 2 aj x2 x +O 6 Z(n) 6 x + O , x j J +1 8 ln log log x x x j=1 n6x 2

Ù¼aj =

Îś(j+1) . 2

y ² : Ă„ky²Ă˜ ÂŞmÂŒĂœŠ:d½n2.8ÂŒ , n = 2k ž, Z(n) = 2n−1, n = pk ž, Z(n) = pk − 1, nÂ?škĂ›ÂƒĂŞĂ?f óêž, Z(n) 6 n − 1; nÂ?šk2‡¹Þ Ă› ÂƒĂŞĂ?f ÛêžkZ(n) 6 (n−1) 2 , Ă?dÂŒ X X X Z(n) = Z(2k) + Z(2k + 1) + O(x) k6[ x2 ]

n6x

6

X

k6[ x2 ]

=3

k6[ x2 ]

2k − 1 +

X

k+

k6[ x2 ]

3 = x2 + O 8

X

k6[ x2 ]

Xp−1

2

p6x

k+

Xp−1 p6x

+

x2 . ln x

X

2

+

X

2k + O(x)

k6ln x

2k + O(x)

k6ln x

e ÂĄ y ² ½ n † ÂŒ Ăœ Š: Ă? Â?P (n)|n Z(n)(Z(n)+1) , kP (n)|Z(n)(Z(n) + 1), ¤ 2

ÂąZ(n) > P (n) − 1,Ă?dÂŒ : X X Z(n) > (P (n) − 1) n6x

n6x

=

X

P (n) + O(x)

n6x

J X aj x2 =x +O . logjx logJx +1 j==1 2

Â

Â˜ĂšĂŤÂ„Šz[25].

2.5 'uSmarandacheŸê†–SmarandacheŸê Â?§I 2.5.1 Âľ9yG ĂŠ?Âż ĂŞnĂ?Âś F. SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m! =S(n) = min{m : m ∈ N, n|m!}. lS(n) ½Ă‚<‚N´íĂ‘XJn = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r

LÂŤn IOŠ)ÂŞ, @oS(n) = max {S(pÎąi i )}. ddĂ˜JOÂŽĂ‘S(1) = 1, S(2) = 16i6r

2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 1, S(12) = 4, S(13) = 13, S(14) = 7, S(15) = 5, S(16) = 6 ¡ ¡ ¡ .'uS(n) 16

1 Ă™ SmarandacheŸê†–SmarandacheŸê

ÂŽâ5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âź Ă˜ k (J[7,27,28,33,36] . ~X, Lu YamingĂŻ Ă„ ˜aÂ?šS(n)Â?§ ÂŒ)5[28] , y² TÂ?§kĂƒÂĄĂľ| ĂŞ), =y² ĂŠ? Âż ĂŞk > 2Â?§ S(m1 + m2 + ¡ ¡ ¡ + mk ) = S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ) kĂƒÂĄĂľ| ĂŞ)(m1 , m2, ¡ ¡ ¡ , mk ).

Jozsef Sandor[27] ?Â˜Ăš`²Ê?Âż ĂŞk > 2, 3ĂƒÂĄĂľ| ĂŞ (m1 , m2 , ¡ ¡ ¡ , mk )

ávĂ˜ ÂŞ S(m1 + m2 + ¡ ¡ ¡ + mk ) > S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ). Ӟ, q 3ĂƒÂĄĂľ| ĂŞ(m1 + m2 + ¡ ¡ ¡ + mk )ávĂ˜ ÂŞ S(m1 + m2 + ¡ ¡ ¡ + mk ) < S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ). d , Mó¸[36] Âź k'S(n) ˜‡ Â?(J[36] . =y² ĂŹCúª ! 3 3 3 X 2 2 2Îś( )x x 2 +O (S(n) − P (n))2 = , 3 ln x ln2 x n6x Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f, Îś(s)LÂŤRiemann Zeta-Ÿê. y3½Ă‚,˜‡ŽâŸêZ(n)Â?

m(m + 1) Z(n) = min m : m ∈ N, n

. 2

TŸêkžÂ? ÂĄÂ?–SmarandacheŸê. 'u§ Ă? 5Â&#x;, Â?,–8 Ă˜Ăľ, ´Â?kĂ˜ <?1LĂŻĂ„, Âź ˜ kdŠ nĂ˜ĂŻĂ„¤J[37−39]. AO3Šz[40] 13Ù¼, Kenichiro KashiharaĂ˜ĂŁ ŸêZ(n) ˜ Ă? 5Â&#x;, ӞÂ?JĂ‘ e¥ß‡ ÂŻK: (A)ÂŚÂ?§Z(n) = S(n) ¤k ĂŞ); (B)ÂŚÂ?§Z(n) + 1 = S(n) ¤k ĂŞ). ! ĂŒÂ‡8 ´|^Ă? Â?{ĂŻĂ„Â?§(A) 9(B) ÂŒ)5, ¿Ÿ Ú߇Â? § ¤k ĂŞ), äN/`, =y² eÂĄ ½n. ½ n 2.11 ĂŠ?Âż ĂŞn > 1,ŸêÂ?§ Z(n) = S(n) ¤å Â…= n = p¡m, Ù¼pÂ?Ă›ÂƒĂŞ, mÂ? p+1 ?ÂżÂŒu1 Ă?ĂŞ. =m| p+1 Â…m > 1. 2 2 17

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

½ n 2.12 ĂŠ?Âż ĂŞn,ŸêÂ?§ Z(n) + 1 = S(n) p−1 ¤å Â…= n = p ¡ m, Ù¼pÂ?Ă›ÂƒĂŞ, mÂ? p−1 2 ?ÂżĂ?ĂŞ. =m| 2 .

w,, T½n”.)Ăť ÂŻK(A)9(B). ½=y² Ú߇Â?§Ă‘kĂƒÂĄĂľÂ‡ ĂŞ), ¿‰Ñ §Â‚z‡) äN/ÂŞ. Ù¼, AO3ÂŤm[1, 100]ÂĽ, Â?§Z(n) = S(n)k9‡), §Â‚ŠO´n = 1, 6, 14, 15, 22, 28, 33, 66, 91.ĂŠuÂŻK(B), w,Â?§Z(n) + 1 = S(n)3ÂŤm[1, 50]ÂĽk19 ‡), §Â‚ŠO´ n = 3, 5, 7, 10, 11, 13, 17, 19, 21, 23, 26, 29, 31, 34, 37, 39, 41, 43, 47. |^Ă? Â?{‰Ñ½n y². Ă„ky²½n2.10. ¯¢Þ, n = 1ž, Â?§Z(n) = S(n)¤å. n = 2, 3, 4, 5ž, w,Ă˜ávÂ?§Z(n) = S(n). u´, b½n > 6ávÂ?§Z(n) = S(n), Ă˜Â” Z(n) = S(n) = k. dŸêZ(n)9S(n) ½Ă‚ÂŒ , k´ ê, ÂŚ náveÂĄ ߇ Ă˜ÂŞ:

k(k + 1) , n|k!. n

2

(2-16)

Ă„k, y²3ÂŞ(2-16)ÂĽk + 1Ă˜ÂŒUÂ?ÂƒĂŞ. ¯¢ÞXJk + 1Â?ÂƒĂŞ, Ă˜Â” k + 1 = p,u´3n| p(p−1) ÂĽ (n, p) = 1ž, ĂĄÂ?Ă­Ă‘n| p−1 .l n Ă˜ 2 2 p−2 X i=1

i=

(p − 1)(p − 2) . 2

چk = p − 1Â? êŒ n| k(k+1) gĂąÂœ (n, p) > 1ž, dupÂ?ÂƒĂŞ, ¤¹ 2

Ă­Ă‘p|n. 2dun|k!, ĂĄÂ? p|k!. Ú´Ă˜ÂŒU , Ă?Â?p = k + 1,¤¹pĂ˜ÂŒU Ă˜(p − 1)!, l y² 3ÂŞ(2-16)ÂĽk + 1Ă˜ÂŒUÂ?ÂƒĂŞ.

Ă™g, y²3ÂŞ(2-16)ÂĽ kÂ?Ûêžk˜½Â?ÂƒĂŞ. ¯¢Þ kÂ?Ûêž k+1 2 Â?

ĂŞ, ekÂ?ĂœĂŞ, K kÂŒ¹Š)¤ßÂ‡Ă˜Ă“ ĂŞ ÂŚĂˆÂž, Ă˜Â” k = a ¡ b, a > 1, b > 1Â…a 6= b. u´5Âż (k, k+1 ) = 1,Ă˜JĂ­Ă‘k = a ¡ b|(k − 1)! k+1 |(k − 1)! 2dun 2 2

Ă˜ k(k+1) ĂĄÂ?Ă­Ă‘n|(k − 1)!. چk´ êŒ n|k!gĂą. kÂ?ĂœĂŞÂ…Â?,˜ 2

ÂƒĂŞ Â?˜ž, k = pÎą .dukÂ?Ûê, ¤¹p > 3, l p, 2p, ¡ ¡ ¡ , pÎąâˆ’1 Ăž uk − 1Â…z

‡êÑ Ă˜(k − 1)!, u´dn Ă˜ k(k+1) E,ÂŒ¹íĂ‘n|(k − 1)!. چk ½Ă‚gĂą, ¤ 2 Âą kÂ?Ă›ĂŞÂžÂ˜½Â?ÂƒĂŞ.

( Ăœ Âą Ăž Ăź ÂŤ Âœ š Ă­ Ă‘ kÂ? Ă› ĂŞ ž kk = p,d žn Ă˜ p(p+1) p!, ´ n 2 p+1 Ă˜ p+1 2 ž,w,kS(n) < p; n = pžZ(n) 6= S(n). ¤¹ÂŒÂą n = p ¡ m, Ù¼m´ 2

18

1 Ă™ SmarandacheŸê†–SmarandacheŸê

?˜Œu1 Ă?ƒ. y3y² n = p ¡ m, Ù¼m´ p+1 ?˜Œu1 Ă?êž, ˜½kZ(n) = S(n).¯¢ 2 p−1 P Ăždžw,kS(pm) = S(p) = p.Ă?Â?mĂ˜ Ă˜ i = p(p−1) , Ă„K†m Ă˜ p+1 gĂą!¤ 2 2 i=1

ÂąZ(pm) = p, l Z(pm) = S(pm). Â

, y²Ă˜ 3óêkÂŚ Z(n) = S(n) = k. ^‡y{5y²Ú˜(Ă˜. b

½ 3 Ăł ĂŞk = 2m, ÂŚ Z(n) = S(n) = k = 2m,K d Âź ĂŞZ(n)9S(n) ½ Ă‚ ÂŒ ,n Ă˜ k(k+1) = m(2m + 1) 9(2m)!. dcÂĄ ŠĂ›ÂŒ 2m + 1Ă˜ÂŒUÂ?ÂƒĂŞ, Ă„K 2 2m−1 P (n, 2m + 1) = 1ž, n Ă˜ i = m(2m − 1), w,چ2m´ êŒ n i=1

Ă˜m(2m + 1)gĂą! (n, 2m + 1) > 1ž,dÂƒĂŞ 5Â&#x;ĂĄÂ?Ă­Ă‘p = 2m + 1|n, l 2

dn|(2m)! p = 2m + 1|(2m)!, gĂą!¤¹2m + 1Ă˜ÂŒUÂ?ÂƒĂŞ, Ă“ ÂŒÂąy²mĂ˜ÂŒ UÂ?ĂœĂŞ, Ă„KN´íĂ‘n|(2m − 1)!, †2m´ êŒ n|(2m)! gĂą!l mÂ? ÂƒĂŞp, k = 2p. u´ÂŒ n|p(2p + 1)! 9n|(2p)!. ´, n up(2p + 1) ?˜Ă?ÂƒÂžĂ‘ ´Ă˜ÂŒU !Â?Ă’´`ĂŠ?Âżk|p(2p + 1), Ă˜ÂŒUkS(k) = 2p. u´, ¤ ½n2.10 y². y3y²½n2.11. †½n2.10 y²Â?{ƒq, ĂšpÂ?Â‰Ă‘ÂŒVL§. b½ ĂŞnávÂ?§Z(n) + 1 = S(n), Âż Z(n) + 1 = S(n) = k. u´dŸêZ(n)9S(n) ½ Ă‚Ă˜JĂ­Ă‘k´ ê, ÂŚ

k(k − 1) n

, n|k!. 2

(2-17)

w ,, ÂŞ f(2-17)ÂĽ kÂ? Ă› ĂŞ ž ˜ ½ Â? ƒ ĂŞ!Ă„ K ÂŒ J Ă‘n|(k − 1)!, †k´ Â

p−1 ĂŞ ÂŚ n|k!g Ăą. Ă? dk = pÂ? ˜ ƒ ĂŞ. 2 dn| p(p−1) 2 , Âż 5 ÂżS( 2 ) < p,ĂĄ Â? Ă­

Ă‘n = p ¡ m,mÂ? p−1 ?˜ Ă?ĂŞ. N´ y n = p ¡ m,mÂ? p−1 ?˜ Ă?êž, ná 2 2 vÂ?§Z(n) + 1 = S(n).

ÂŞ(2-17)ÂĽk = 2mÂ?óêž, k − 1 = 2m − 1˜½Â?ÂƒĂŞ, l ÂŒ Ă˜ 3Ăš

ĂŞnÂŚ Z(n) + 1 = S(n) = 2m. ¤¹, Â?§Z(n) + 1 = S(n) ¤å Â…= n = p ¡ m, Ù¼mÂ? p−1 2 ?˜ Ă?ĂŞ. u´, ¤ ½n2.11 y².

2.5.2 ˜ ƒ'ÂŻK 'uSmarandacheŸêS(n)9–SmarandacheŸêZ(n)5Â&#x; ĂŻĂ„Â?, Ă˜ ?Ă?, ´E, 3Ă˜ ÂŻK. Â? Buk, Ă–Ăś?1ĂŤ Ăš?Â˜ĂšĂŻĂ„, Ăšp0 ˜ †ŸêS(n)9Z(n) k' ¿…k¿Â ÂŻK. ÂŻ K 2.1 ĂŠ?Âż ĂŞn, H(n)LÂŤÂŤm[1, n]¼¤kÂŚS(n)Â?ÂƒĂŞ ĂŞ ‡

19

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

ĂŞ. Ă ĂŻĂ„H(n) ĂŹC5Â&#x;. Ă&#x;Ăż: H(n) = 1. n→∞ n lim

ÂŻ K 2.2  þkk Â‡ ĂŞn,ÂŚ X 1 S(d) d|n

� ê, Ù¼ ¯ K 2.3

P

d|n LÂŤĂŠn ¤k Ă?ĂŞÂŚĂš. ?Â˜ĂšĂ&#x;ÿÞªÂ? ĂŞ Â…= n P (n ĂŻĂ„Ÿê PS(n) (n) 9 S(n) ) ފ5Â&#x;, ¿‰Ñފ

= 1, 8.

X S(n) X P (n) , P (n) S(n)

n6x

n6x

Â˜Â‡ĂŹCúª, Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f. xÂŞuĂƒÂĄÂž, Ă&#x;Ăż1Â˜Â‡ĂžÂŠĂŹC uc ¡ x,Ù¼cÂ?,˜Œu1 ~ĂŞ;1 ‡ÞŠAT†xĂ“ . ,ÂŻK2.3†¯K2.1—ƒ ƒ'. XJÂŻK2.1ÂĽ Ă&#x;Ăż (, @oĂ’ÂŒ¹Ÿ 1 ‡ÞŠ ĂŹCúª. ÂŻ K 2.4 ŸêZ(n) ŠŠĂ™ĂŠĂ˜5K, ĂŠk nXn =

m(m+1) , 2

kZ(n) = m <

√ 2n.

ĂŠu,˜ nXn = 2Îą , kZ(n) = 2Îą+1 − 1 = 2n − 1. Ă?dk7‡ïÄZ(n) ފ5 Â&#x;, ‰Ñފ

X

n6x

Z(n),

X

ln(Z(n)),

n6x

X

n6x

1 Z(n)

Â˜Â‡ĂŹCúª. ÂŻ K 2.5 ÂŚÂ?§Z(n) = φ(n) ¤k ĂŞ), Ù¼φ(n)Â?EulerŸê. ĂšÂ˜Â?§k Ăƒ þ‡ ĂŞ), ~XnÂ?ÂƒĂŞpžÞávÂ?§. n = 2pÂ…p ≥ 1(mod4)ž, nÂ?áv

TÂ?§. Ă˜ Ăš ²Â…) , ´Ă„„kٌ ĂŞ)´Â˜Â‡Ăşm ÂŻK. Ă&#x;ĂżTÂ?§Â? kn = 1Âą9Þãߍ). ÂŻ K 2.6 ÂŚÂ?§S(Z(n)) = Z(S(n)) ¤k ĂŞ). Ă&#x;ĂżTÂ?§ þkk Â‡ ĂŞ).

2.6 'uSmarandacheŸê†–SmarandacheŸê Â?§II ĂŠ?Âż ĂŞn,Ă?Âś F.SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!, =S(n) = min{m : m ∈ N, n|m!}, d Âź ĂŞ ´ Ă? Âś ĂŞ Ă˜ ; [F.Smarandache3 5Only

Problems,Not Solutions6Â˜Ă–ÂĽĂš\ , ¿ïÆ<╀§ 5Â&#x;. lS(n) ½Ă‚N ´íĂ‘:XJn = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r LÂŤn IOŠ)ÂŞ, @oS(n) = max {S(pÎąi i )}.ddÂ? 16i6r

Ă˜JOÂŽĂ‘S(n) cA‡ŠÂ?:S(1) = 1, S(2) = 2S(4) = 4, S(5) = 5, S(6) = 3, S(7) =

20

1 Ă™ SmarandacheŸê†–SmarandacheŸê

7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 11, S(12) = 4, S(13) = 13, S(14) = 7, S(15) = 5, S(16) = 6, ¡ ¡ ¡ .'uS(n) ÂˆÂŤÂŽâ5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âź Ă˜ k ( J, ĂŤ Šz[7,27,28,33] . ~X3Šz[36]ÂĽ, ĂŻĂ„ Úª X 1 d|n

(2-18)

S(d)

Â? ĂŞ ÂŻK, Âży² eÂĄ3‡(Ă˜: (a) nÂ?Ăƒ²Â?Ă?fêž, (2-18)ÂŞĂ˜ÂŒU´ ĂŞ; (b)ĂŠ?ÂżĂ›ÂƒĂŞp9?Âż ĂŞÎą, n = pÎą Â…Îą 6 pž, (2-18)ÂŞĂ˜ÂŒU´ ĂŞ; Îą

k−1 (c)ĂŠu?Âż ĂŞn, n IOŠ)ÂŞÂ?pÎą1 1 ¡ pÎą2 2 , ¡ ¡ ¡ , pk−1 ¡ pk Â…S(n) = pk ž,

(2-18)ÂŞĂ˜ÂŒU´ ĂŞ.

d , 3Šz[27]ÂĽ, J. SandorĂš\ –F. SmarandacheŸêZ(n)Xe: Z(n)½Ă‚Â? ,=  êm, ÂŚ n Ă˜ m(m+1) 2

m(m + 1) Z(n) = min m : m ∈ N, n

. 2

lZ(n) ½ Ă‚ N ´ Ă­ Ă‘Z(n) c A ‡ Š Â?: Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 8, Z(10) = 4, Z(11) = 10, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 31, ¡ ¡ ¡ .'uZ(n) ÂŽâ5Â&#x;, NþÆÜÂ?

?1 ĂŻĂ„, Âź Ă˜ k (J, ĂŤ Šz[38 − 41]. ! ĂŒÂ‡8 ´ïĂ„Ÿê Â?§

Z(n) + S(n) = kn

(2-19)

ÂŒ)5, Ù¼kÂ??Âż ĂŞ, Âż|^Ă? 9|ĂœÂ?{Âź ĂšÂ˜Â?§ ¤k ĂŞ ). äN/`Â?Ă’´y² eÂĄ ½n: ½ n 2.13 k = 1ž, n = 6, 12´Â?§(2-19)=k ߇AĂ? ĂŞ); džÙ§ ĂŞnávÂ?§(2-19) Â…= n = p ¡ u½Ün = p ¡ 2Îą ¡ u, Ù¼p > 7Â?ÂƒĂŞ, 2Îą |p − 1,

u´ p−1 ?ÂżÂ˜Â‡ÂŒu1 ÛêĂ?f. 2Îą

½ n 2.14 k = 2ž, n = 1´Â?§(2-19) ˜‡AĂ?); Ă™§ ĂŞnávÂ?§(2-19) Â…= n = p ¡ u,Ù¼p > 5Â?ÂƒĂŞ,u´ p−1 2 ?ÂżÂ˜Â‡óêĂ?f. 5Âż , Z(n) 6 2n − 19S(n) 6 n, ¤¹ k > 2ž, Â?§(2-19)vk ĂŞ). l n

½nĂŠN´ÊÂŽ FermatÂƒĂŞ, =/XÂŞFn = 22 + 1 ÂƒĂŞ, Ù¼n > 1Â? ĂŞ. ~ XF1 = 5, F2 = 17, F3 = 257 . d½n2.13Ă˜JĂ­Ă‘eÂĄ Ă­Ă˜: Ă­ Ă˜ 2.1 k = 1ž, XJnškFermatƒĂ?f, KnĂ˜ÂŒUávÂ?§(2-19).

21

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

|^Ă? 9|ĂœÂ?{5 ¤½n y². Ă„ky²½n2.13. ڞk = 1. 5Âż Z(1) + S(1) = 2 6= 1, Z(2) + S(2) = 5 6= 3, Z(3) + S(3) = 5 6= 3, Z(4) + S(4) = 11 6=

4, Z(5) + S(5) = 9 6= 5, Z(6) + S(6) = 6,¤¹n = 1, ¡ ¡ ¡ , 5Ă˜ávÂ?§(2-19). n = 6áv Â?§(2-19). u´ Ă™§návÂ?§(2)ž˜½kn > 7. n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k Â?n IOŠ )ÂŞ, dždF.SmarandacheŸê 5Â&#x;

S(n) = max {S(pÎąi i )} ≥ S(pÎą ) = u ¡ p, 16i6r

Ù¼pÂ?,˜pi , ÎąÂ?,Â˜Îąi , u 6 Îą. y35Âż p|n9S(n) = u ¡ p,¤¹ÂŒ n = pÎą ¡ n1 , návÂ?§(2-19)žk Z(n) + u ¡ p = pÎą ¡ n1 .

(2-20)

Ă„ky²3(2-20)ÂŞÂĽÎą = 1. Ă„Kb½ι > 2, u´d(2-20)ÂŞĂĄÂ?Ă­Ă‘p|Z(n) = m.dZ(n) = m ½ Ă‚ n = pÎą ¡ n1 Ă˜ m(m+1) , (m, m + 1) = 1,¤ ÂąpÎą |m.l 2

d(2-20)ÂŞĂ­Ă‘pÎą |S(n) = u ¡ p, =pÎąâˆ’1 |u,l pÎąâˆ’1 6 u. ´,˜Â?ÂĄ, 5Âż S(n) = S(pÎą ) = u ¡ p, dF.SmarandacheŸêS(n) 5Â&#x; u 6 Îą,¤¹pÎąâˆ’1 6 u 6 Îą. dÂŞ

ĂŠ Ă› ƒ ĂŞpw , Ă˜ ¤ ĂĄ. X Jp = 2, K Îą > 3ž, pÎąâˆ’1 6 u 6 Îą Â? Ă˜ ¤ ĂĄ. u ´ Â? k ˜ ÂŤ ÂŒ U:u = a = 2. 5 Âż n > 5Âą 9S(n) = 4,¤ Âą d ž Â? k ˜ ÂŤ ÂŒ U:n = 12, n = 12´Â?§(2) ˜‡). ¤¹XJĂ™§ ĂŞnávÂ?§(2-19),K(220)ÂŞÂĽ7kS(n) = p, Îą = u = 1.3ڍÂœše, -Z(n) = m = p ¡ v, K(2-20)ª¤

Â?

v + 1 = n1 , ½Ün1 = v + 1,=n = p ¡ (v + 1), Z(n) = p ¡ v. 2dZ(n) ½Ă‚ n = p ¡ (v + 1) Ă˜ Z(n)(Z(n) + 1) pv ¡ (pv + 1) = , 2 2 ½Ü(v + 1) Ă˜ Z(n)(Z(n) + 1) v ¡ (pv + 1) = , 2 2 5Âż (v + 1, v) = 1,¤¹ vÂ?óêždÞªåÂ?Ă­Ă‘v + 1|pv + p − p + 1,=v +

p−1 1|p − 1½Üv + 1| p−1 2 . w,ĂŠ 2 ?ÂżÂŒu1 ÛêĂ?fr, n = p ¡ r ´Â?§(2-19) ).

Ă?Â?džkZ(p ¡ r) = p ¡ (r − 1).

vÂ?Ûêž, d(v + 1) Ă˜ Z(n)(Z(n) + 1) v ¡ (pv + 1) = , 2 2

22

1 Ă™ SmarandacheŸê†–SmarandacheŸê

ddÂŒĂ­Ă‘

pv + 1 (p − 1)(v + 1) + v − p + 2 (v + 1)

= . 2 2

p − 1 = (2k + 1) ¡ (v + 1). u´ p − 1 = 2β ¡ h,Ù¼hÂ?Ûê, K v+1 Â? uh ÛêĂ?f. N´ yĂŠ?¿Û 2β ĂŞr|hÂ…r < h, n = p ¡ 2β ¡ r Â?Â?§(2-19) ). Ă?Â?džk Z(p ¡ 2β ¡ r) = p ¡ (2β ¡ r − 1). ¯¢Þ, 5Âż r|h. Ă„kN´íĂ‘p ¡ 2β ¡ r Ă˜ m(m+1) .u´dZ(n) ½Ă‚ 2 Z(p ¡ 2β ¡ r) = p ¡ (2β ¡ r − 1). u´y² ½n2.13. y3y²½n2.14. dž5Âż k = 2, ¤¹ n = 1ž, kZ(1) + S(1) = 2=n = 1´ Â?§(2-19) ˜‡). XJÂ?§(2-19)„kĂ™§ ĂŞ)n > 2,Kd½n2.13 y²Â?{ Ă˜JĂ­Ă‘n = p ¡ u, Ù¼p > 5Â?ÂƒĂŞ, S(u) < p. “\Â?§(2-19)ÂŒ Z(p ¡ u) + S(p ¡ u) = 2p ¡ u. ddÂŞĂĄÂ?Ă­Ă‘p Ă˜Z(p ¡ u). Z(p ¡ u) = p ¡ v, Kv = 2u − 1.dZ(n) ½Ă‚ p ¡ u p−1 Ă˜ p(2u−1)(p(2u−1)+1) . l u Ă˜ p−1 2 2 . d , uÂ? 2 ?˜Œu1 ÛêĂ?êž, Z(p ¡

u) = p ¡ (u − 1), ¤¹džn = p ¡ uĂ˜´Â?§(2-19) ĂŞ); uÂ? p−1 2 ?˜óêĂ? êžk

Z(p ¡ u) = p ¡ (2u − 1), dž Z(p ¡ u) + S(p ¡ u) = 2p ¡ u. u´ ¤ ½n2.14 y². d½n2.13Ă˜JĂ­Ă‘Š¼ Ă­Ă˜. ¯¢Þ½n2.13 ÂĽ ÂƒĂŞĂ˜ÂŒU´FermatÂƒĂŞ, Ă?Â? pÂ?Fermat ÂƒĂŞÂž, p − 1vkÂŒu1 ÛêĂ?f.

2.7 'uSmarandacheŸêÂ†ÂƒĂŞ ĂŠ?Âż ĂŞn, Ă?Âś F.SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n | m!. 23

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

= Ă’ ´S(n) = min{m : m ∈ N, n|m!}. Ăš ˜ Âź ĂŞ ´ { 7 Ă› ĂŞ Z ĂŚ Ă? Âś ĂŞ Ă˜ ; [F.Smarandache Ç3Œ¤Ă? 5Only Problems, Not Solutions6Â˜Ă–ÂĽĂš\ , Âż ïÆ<╀§ 5Â&#x;ÂœlS(n) ½Ă‚<‚N´íĂ‘XJn = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r LÂŤn I

OŠ)ÂŞ, @oS(n) = max {S(pÎąi i )}. ddÂ?Ă˜JOÂŽĂ‘S(n) cA‡ŠÂ?: S(1) = 1, 16i6r

S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 11, S(12) = 4, S(13) = 13, S(14) = 7, S(15) = 5, S(16) = 6, ¡ ¡ ¡ ¡ ¡ ¡ . 'uS(n) ÂŽâ5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âź Ă˜ k (JÂœĂŤ Š

z[28, 32, 36, 37, 57]. ~X, Lu Yaming[28] 9Wju[32] ĂŻĂ„ ˜aÂ?šS(n)Â?§ ÂŒ)

5, y² TÂ?§kĂƒÂĄĂľ| ĂŞ). =Ă’´y² ĂŠ?Âż ĂŞk > 2, Â?§ S(m1 + m2 + ¡ ¡ ¡ + mk ) = S(m1 ) + S(m2 ) + ¡ ¡ ¡ + S(mk ) kĂƒÂĄĂľ| ĂŞ)(m1 , m2 , ¡ ¡ ¡ , mk ).

Mó¸[36] Âź k'S(n) ˜‡ Â?(JÂœÂ?Ă’´y² ĂŹCúª ! 3 3 3 X 2 2 x 2Îś x 2 +O (S(n) − P (n))2 = , 2 3 ln x ln x n6x

Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f, Îś(s)LÂŤRiemann zeta-Ÿê. ÚÂ=[57] ĂŻĂ„ 'uSmarandacheŸê ˜‡Ă&#x;ÂŽ, =Ă’´y² nÂ?, AĂ? ĂŞ(~XnÂ?Ăƒ²Â?Ă?fĂŞ)ž, Úª X 1 S(d) d|n

Ă˜ÂŒUÂ? ĂŞ. y3-PS(n)LÂŤÂŤm[1, n]ÂĽS(n)Â?ÂƒĂŞ ĂŞn ‡ê. 3˜Â&#x;™uL Šz PS(n) ÂĽ, J.CastilloïÆïÄ4 lim 3ÂŻK. XJ 3, (½Ă™4 . ĂšÂ˜ÂŻK n−→∞ n ´k , § ÂŤ F.SmarandacheŸêS(n)ŠŠĂ™ 5Æ5, ӞÂ?`²3Ă˝ÂŒþê Âœše, F.SmarandacheŸêS(n) ÂƒĂŞÂŠ! , 'uĂšÂ˜ÂŻK, duĂ˜ lĂ›eĂƒ, ¤¹Â–8vk<ĂŻĂ„, – vkw L k'Â?ÂĄ Ă˜Š. ! ĂŒÂ‡8 ´|^Ă? Â?{ĂŻĂ„ĂšÂ˜ÂŻK, Âż ”.)ĂťÂœä N/`Â?Ă’´y² eÂĄ : ½ n 2.15 ĂŠ?Âż ĂŞn > 1, kĂŹCúª PS(n) =1+O n

1 . ln n

w,ڴ˜‡'J.CastilloÂŻKÂ?r (Ă˜. ,XĂ›U?Þª¼ Ă˜ ‘Â?´Â˜ ‡k¿Â ÂŻK, k–u?Â˜ĂšĂŻĂ„! dd½nĂĄÂ? eÂĄ : 24

1 Ă™ SmarandacheŸê†–SmarandacheŸê

Ă­ Ă˜ 2.2 ĂŠ?Âż ĂŞn, k4 PS(n) = 1. n−→∞ n lim

Ăš!|^Ă? Â?{‰Ñ½n † y². Ă„k On − PS(n) Ăž.. ¯¢Þ

n > 1ž, n = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r LÂŤn IOŠ)ÂŞ, @odŸêS(n) ½Ă‚95Â&#x;ÂŒ

S(n) = S (pÎąi i ) = m ¡ pi . eÎąi = 1, @om = 1Â…S(n) = pi Â?ÂƒĂŞ. eÎąi > 1, @ om > 1, KS(n)Â?ĂœĂŞ. ¤¹n − PS(n)Â?ÂŤm[1, n]¼¤kS(n) = 19S(n)Â?ĂœĂŞ

n ‡ê! w,S(n) = 1 Â…= n = 1. u´-M = ln n, Kk X X X 161+ 1+ n − PS(n) = 1 + k6n S(k)=S(pÎą ), Îą>2

y3ŠO OÂŞ(2-21)ÂĽ ˆ‘, w,k X X X 16 1+ kpÎą 6n Îąp>M, Îą>2

≪

h

M 2

X

√ <p6 n

n + p2

pÎą 6n Îąp>M, Îą>3

n n ≪ + ι p ln n

≪

X n n + + Îą ln n √ p

≪

n n n n + √ + ≪ . M −1 ln n 2 M ln n

p6 n √ Îą> M

X

√ p6 n √ p> M , Îą>3

X

√ M <p6 n 2

kpÎą 6n Îąp>M, Îą>3

kp2 6n 2p>M

X

16

X

√ p6 n Îąp>M, Îą>p

X

1.

(2-21)

kpÎą 6n Îąp>M, Îą>2

S(k)6M

1+

k6 pn2

n + pÎą

X

X

1

pÎą 6n k6 pnÎą Îąp>M, Îą>3

X

√ p6 n Îąp>M, 36Îą<p

n pÎą

n pÎą (2-22)

ĂŠuÂŞ(2-21)ÂĽ ,˜‘, I‡Ì # OÂ?{. ĂŠ?ÂżÂƒĂŞp 6 M , -Îą(p) = i Y M M , =Ă’´ι(p)LÂŤĂ˜Â‡L  ÂŒ ĂŞ. u = pÎą(p) . ĂŠ?¿ávS(k) 6 p−1 p−1 p6M

M ĂŞk, S(k) = S(pÎą ), KdS(k) ½Ă‚˜½kpÎą |M !, l Îą 6

∞ X M j=1

pj

6

M . ¤¹¤kávS(k) 6 M ĂŞk˜½ Ă˜u, Ăš k Â‡ĂŞĂ˜ÂŹÂ‡Lu Ă? p−1 ĂŞ ‡ê, =Ă’´d(u). ¤¹k X X Y 16 1= (1 + Îą(p)) S(k)6M

p6M

d|u

=

Y

p6M



= exp 

M 1+ p−1 X

p6M

 M , ln 1 + p−1

25

(2-23)

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ù¼exp(y) = ey . dƒê½n ߍ/ÂŞ(ĂŤ Šz[4]9[6])

Ď€(M ) =

X

p6M

M 1= +O ln M

M ln2 M

,

X

ln p = M + O

p6M

M ln M

ÂŒ : X M M ln 1 + 6 ln 1 + p−1 p−1 p6M p6M X 1 = ln (p − 1 + M ) − ln p − ln 1 − p p6M X X 1 6 Ď€(M ) ¡ ln(2M ) − ln p + p p6M p6M M M M ¡ ln(2M ) =O . −M +O = ln M ln M ln M X

(2-24)

5Âż M = ln n, dÂŞ(2-23)9ÂŞ(2-24)ĂĄÂ? OÂŞ:

X

S(k)6M

c ¡ ln n 1 ≪ exp , ln ln n

(2-25)

Ù¼cÂ?˜ ~ĂŞ. 5Âż exp

c¡ln n ln ln n

n , ln n

u´(ĂœÂŞ(2-21), ÂŞ(2-22)9ÂŞ(2-25)ĂĄÂ?Ă­Ă‘ OÂŞ: n X 1=O . n − PS(n) = 1 + ln n ≪

k6n S(k)=S(pÎą ), Îą>2

¤¹ PS(n) = n + O u´ ¤ ½n y².

26

n . ln n

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜

Ă™§Ÿê 3.1 ˜a†–SmarandacheŸêƒ' ŸêÂ?§ c¥‰Ñ SmarandacheŸêS(n), Ă™½Ă‚Â? êmÂŚ n|m!, =S(n) = min{m : m ∈ N, n|m!}. lS(n) ½Ă‚N´íĂ‘: XJn = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k LÂŤn IO Š)ÂŞ, @o

S(n) = max{S(pÎą1 1 ), S(pÎą2 2 ), ¡ ¡ ¡ , S(pÎąk k )}. ' u Âź ĂŞS(n) ÂŽ â 5 Â&#x;, ĂŤ Š z[7, 36, 53 − 55, 57]. 3 Š z[58]ÂĽ, SandorĂš \

SmarandacheŸêS(n) ÊóŸêS ∗ (n), ½Ă‚Xe, ĂŠu?Âż ĂŞn, S ∗ (n)½Ă‚Â?

 ÂŒ ĂŞmÂŚ m!|n, =k S ∗ (n) = max{m : m ∈ N, m!|n}. 'uS ∗ (n) ÂŽâ5Â&#x;Â?kÆÜ?1LĂŻĂ„, ˜X ĂŻĂ„¤J. ~X3Šz[59]ÂĽ ZĂŻĂ„ k'S ∗ (n) ŸêÂ?§ X

SL∗ (d) =

d|n

X

S ∗ (d)

d|n

ÂŒ)5Âż ˜‡k (Ă˜, =e    X  X A= n: SL∗ (d) = S ∗ (d), n ∈ N ,   d|n

KÊu?¿ ¢ês, Drichlet?êf (s) =

d|n

P

n>1,n∈A

kĂ° ÂŞ

1 3s ns

6 1žuÑ, 3s > 1žÂù, …

1 f (s) = Îś(s) 1 − s , 12

Ù¼SL∗ (n)Â?Smarandache LCM ÊóŸê, Â…½Ă‚Â? SL∗ (n) = max{k : [1, 2, ¡ ¡ ¡ , k]|n, k ∈ N}, Îś(s)LÂŤRiemann zetaŸê. 3Šz[27], SandorĂš\ –SmarandacheŸêZ(n)½Ă‚Xe: ĂŠu?Âż ĂŞn,

27

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Z(n)� êm, Œ n| m(m+1) ,= 2

m(m + 1)

Z(n) = min m : m ∈ N, n

. 2

lZ(n) ½Ă‚ÂŒÂąOÂŽĂ‘Z(n) cA‡ŠÂ?: Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 8, Z(10) = 1, ¡ ¡ ¡ . 'uZ(n) ÂŽâ5 Â&#x;, NþÆÜÑ?1 ĂŻĂ„, Âź Ă˜ k¿Â (J[4,6,63−68] , Ӟ Z(n)˜ {

Ăź5Â&#x;: a)ĂŠu?Âż ĂŞÎą9Ă›ÂƒĂŞp, Z(pÎą ) = pÎą − 1; b)ĂŠu?Âż ĂŞÎą,Z(2Îą ) = 2Îą+1 − 1.

!ĂŒÂ‡0 'u–SmarandacheŸê†SmarandacheÊóŸê ŸêÂ?§ Z(n) + S ∗ (n) − 1 = kn,

k>1

(3-1)

ÂŒ)5. ڇ¯K8c„vk<ĂŻĂ„, ĂœŠ+ Ă‡ĂŻĂ†ĂŻĂ„ĂšÂ˜aÂ?§ ĂŞ) Âœ š, ÚÊ?Â˜ĂšĂŻĂ„ŸêS ∗ (n)†Z(n) 5Â&#x;9§Â‚ƒm 'Xk˜½nĂ˜Ă„:,  ­ ‡ ´ÂŒÂą)݆§Âƒ' ?ĂŞ ùÑ5 ÂŻK, ˜ Ă? AĂ?(J, l WĂ– ĂšÂ˜ĂŻĂ„+Â? ˜x. !|^Ă? 9|Ăœ Â?{Âź ڇÂ?§ ¤k ĂŞ). ½ =y² eÂĄ ½n: ½ n 3.1 ŸêÂ?§(3-1) k = 1ž, Â…= Â?kÂ?˜ )n = 1; k = 2ž, Â…= n = 2Îą , Îą > 1ávÂ?§(3-1); k > 3ž, Â?§(3-1)Ăƒ). |^Ă? 9|Ăœ Â?{ÂŒÂąÂ† ‰Ñ½n y². Â? {ßü„, Ă˜Â” S ∗ (n) = m. k = 1ž, w,n = 1ávÂ?§(3-1). ¤¹n = 1´Â?§(3-1) ˜‡). eÂĄb½n > 1Â…ávÂ?§(3-1). d–SmarandacheŸêZ(n) ½Ă‚ Z(n)(Z(n) − 1) + mZ(n) = nZ(n). d d ÂŞ Âż ( ĂœZ(n) ½ Ă‚ ĂĄ Â? ÂŒ Âą Ă­ Ă‘n Ă˜mZ(n), 5 Âż m!|n, ¤ Âą ÂŒ Âą mZ(n) = qn, ½ÜZ(n) =

qn . m

òdª“<ÂŞ(3-1)ÂŒ qn + m − 1 = n. dS ∗ (n) = m

m ½Ă‚ m!|n, l ÂŒ n = m! ¡ n1 , dž, ުŒzÂ?

q(m − 1)! ¡ n1 + m − 1 = m! ¡ n1 .

(3-2)

3 ÂŞ(3-2)ÂĽ k Ăź ‘ w , ÂŒ Âą (m − 1)! Ă˜, ¤ Âą d Ă˜ 5 Â&#x; ´ ÂŞ(3-2)ÂĽ

1 n ‘m − 1Â? U (m − 1)! Ă˜. w ,(m − 1)! Ă˜m − 1 Â… = m = 1, 2, 3.

m = 1ž, dÂŞ(3-2)ÂŒ q = 1, džkZ(n) = n, dŸêZ(n) ½Ă‚95Â&#x;ÂŒ ,

vk ĂŞn > 1ávZ(n) = n. m = 2ž, dÂŞ(3-1) Z(n) = n − 1Â…n > 1, d

žn = pÎą , Îą > 1, pÂ?Ă›ÂƒĂŞ, ²u n = pÎą , Îą > 1Ă˜´ª(3-1) ). m = 3ž, 28

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

=S ∗ (n) = m = 3, Kn = 6, ² yn = 6Ă˜ávÂŞ(3-1).¤¹ k = 1ž, ĂŞnáv ÂŞ(3-1) Â…= n = 1. k = 2ž, w,n = 1Ă˜ávÂŞ(3-1). Ă“n, b½n > 1Â…ávÂ?§(3-1). ŠâZ(n) ½Ă‚ džk Z(n)(Z(n) − 1) + mZ(n) = 2nZ(n). ĂĄÂ?ÂŒ¹ín Ă˜mZ(n), 5Âż m!|n, ÂŒÂą mZ(n) = q ′ n, ½ÜZ(n) =

q′ n m .

òd

q′ n

ª“\ÂŞ(3-1)ÂŒ m +m−1 = 2n. dS ∗ (n) = m ½Ă‚ m!|n, l ÂŒ n = m!¡n2 , džުŒzÂ? k(m − 1)! ¡ n2 + m − 1 = 2m! ¡ n2 .

(3-3)

3 ÂŞ(3-3)ÂĽ k Ăź ‘ w , ÂŒ Âą (m − 1)! Ă˜, ¤ Âą d Ă˜ 5 Â&#x; ´ ÂŞ(3-3)ÂĽ

1 n ‘m − 1Â? U (m − 1)! Ă˜. w ,(m − 1)! Ă˜m − 1 Â… = m = 1, 2, 3.

m = 1ž, d(3-3)ÂŞÂŒ k = 2, džkZ(n) = 2n, dŸêZ(n) ½Ă‚95Â&#x;ÂŒ ,

vk ĂŞn > 1 ávZ(n) = 2n. m = 2ž,d(3-1)ÂŞ Z(n) = 2n − 1Â…n > 1,¤

Âą,n = 2Îą , Îą > 1´ª(3-1) ). m = 3ž,=S ∗ (n) = m = 3,Kn = 6, ² yn = 6Ă˜ áv(3-1)ÂŞ. nĂž, k = 2ž, ĂŞnáv(3-1)ÂŞ Â…= n = 2Îą , Îą > 1. k > 3ž, d¹ÞߍÂœš y²Ă“nÂŒ¹í , vk ĂŞn > 1ávZ(n) = kn,ĂšZ(n) = kn − 1, Â…n = 6Â?Ă˜áv(3-1)ÂŞ. Â?§(3-1)Ăƒ). nÞ¤ã, B ¤ ½n3.1 y².

3.2 'uSmarandachep‡Ÿê†–SmarandacheŸê Â?§ Ă?Âś SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!,=S(n) = min{m : m P n|m!}. Ă? Âś –SmarandacheÂź ĂŞZ(n)½ Ă‚ Â? á v kU n Ă˜  k=1

ĂŞm,=Z(n) = min{m : n|(m(m + 1))/2}.~X,Z(n) cA‡ŠÂ?Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 5, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 9, Z(10) = 4, Z(11) = 10, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 3, Z(17) = 16, Z(18) = 8, Z(19) = 18, Z(20) = 15, ¡ ¡ ¡ .

' u Âź ĂŞS(n)ĂšZ(n), N Ăľ Æ Ăś ĂŻ Ă„ § ‚ 5 Â&#x;, Âż ˜ ­ ‡ (

J[7,29,69−72] . 32.5!ĂŻĂ„ Â?§ Z(n) = S(n),

Z(n) + 1 = S(n)

ÂŒ)5, ¿‰Ñ Â?§ Ăœ ĂŞ).

29

Smarandache ¼ê9Ù '¯KïÄ

©z[75] Ú? Ͷ Smarandachep ¼êSc(n),½Â max{m : y|n!, 1 < y 6 m, (m + 1)†n!}. ~X, Sc(n) cA Sc(1) = 1, Sc(2) = 2, Sc(3) = 3, Sc(4) = 4, Sc(5) = 6, Sc(5) = 5, Sc(7) = 10, Sc(8) = 10, Sc(9) = 10, Sc(10) = 10, Sc(11) = 12, Sc(13) = 16, Sc(14) = 16, Sc(15) = 16, · · · . ©z[72] Ó ïÄ Sc(n)¼êÚZ(n)¼ê m 'X § Z(n) + Sc(n) = 2n, ­ (J, ¿JÑ e¡ ß . ß : én ∈ N + , §

Sc(n) + Z(n) = 2n

¤á = n = 1, 3α , p2β+1, α ¦ 3α + 2 ê u u2 ê, p > 5 ê,β ¦ p2β+1 + 2 ê ? ê. !ò Ñù ß y², = e¡ ½n. ½ n 3.2 n´Û ê , § Sc(n) + Z(n) = 2n ) U n = 1, 3α , p2β+1 , α ¦ 3α + 2 ê u u2 ê, p > 5 ê, β ¦ p2β+1 + 2 ê ? ê. ½ n 3.3 n´óê ,©ü« ¹: (1)n 2 , nØ´ §Sc(n) + Z(n) = 2n ); (2)n´¿© óê, n k3 ØÓ Ïf , nØ´ §Sc(n) + Z(n) = 2n ). y²½n3.2Ú½n3.3, k Ñe¡A Ún. Ú n 3.1 eSc(n) = x ∈ N + , n 6= 3,Kx + 1 un ê. Ú n 3.2 (1)eα ∈ N + ,KZ(2α ) = 2α+1 − 1;

(2)ep´Ø u2 ê, α ∈ N + , KZ(pα ) = pα − 1.

Ú n 3.3 n ∈ N + , n > 59 , KkSc(n) < 3n/2. Ú n 3.4 ên¿© , 3n n + n7/12 m7k ê. Ún3.1-3.4 y²ë ©z[30, 75, 77]. e¡y²½n3.2. n´Û ê , ©6« ¹5?Ø. 30

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

(1)n = 1÷v §. (2)n = 3, Z(3) = 2, Sc(3) = 3, 3Ø´ § ). (3)n = 3α (α > 2), 3α + 2 ê, K Sc(3α ) = 3α + 1,

Z(3α ) = 3α − 1,

3α ´ § ). (4)n = 3α (α > 2), 3α + 2Ø ê, KdÚn3.1 Ún3.2, k Sc(3α ) > 3α + 1,

Z(3α ) = 3α − 1,

3α Ø´ § ). (5)n = pγ , γ > 1, p > 5 ê, K Z(pγ ) = pγ − 1 n 6= 3, ù ©2« ¹:

en = p2β , β > 1, Kp ≡ ±1(mod3),u´ p2β ≡ 1(mod3),

l p2β + 2 ≡ 0(mod3), Kp2β + 2 Ø U ê, dÚn3.1, d nØ´ § ). en = p2β+1 ,Ù¥p > 5 ê, β ¦ p2β+1 + 2 ê ?¿ ê, dÚ n3.1, Sc(p2β+1 ) = p2β+1 + 1. dÚn3.2, Z(p2β+1 ) = p2β+1 − 1,KSc(p2β+1 ) + Z(p2β+1 ) = p2β+1 + 1 + p2β+1 − 1 = 2p2β+1 , n = p2β+1 §¤á. ep2β+1 + 2Ø ê, KdÚn3.1, Sc(p2β+1 ) > p2β+1 , n = p2β+1 Ø´ § ). (6)2†n, n = pα1 1 n1 , (p1 , n1 ) = 1, α1 > 1, p1 > 3 ê, Ó{ § n1 x ≡ 1(modpα1 1 ) k), ? Ó{ § n21 x2 ≡ 1(modpα1 1 ) k), Ù)Ø y,K 1 6 y 6 pα1 1 − 1,qpα1 1 − y½ c¡Ó{ § ), K

31

Smarandache ยผรช9ร ย 'ยฏKรฏร

1 6 y 6

ฮฑ

(p1 1 โ 1) . 2

d n21 y 2 โ ก (modpฮฑ1 1 ),

K pฮฑ1 1 |(n1 y โ 1)(n1 y + 1), (n1 y โ 1, n1 y + 1)|2, uยด(pฮฑ1 1 โ 1)|(n1 y โ 1)ยฝ(pฮฑ1 1 โ 1)|(n1 y + 1).

ฮฑ1 ฮฑ1 1 y) e(p1 โ 1)|(n1 y โ 1),Kn = p1 n2 (n1 yโ 1)(n , l 2 e(pฮฑ1 1

(pฮฑ1 โ 1)n1 n Z(n) = m 6 n1 y โ 1 6 1 โ 16 , 2 2

1 y) โ 1)|(n1 y + 1), Kn = pฮฑ1 1 n1 (n1 yโ 1)(n , l 2 Z(n) = m 6 n1 y 6

n (pฮฑ1 1 โ 1)n1 6 . 2 2

dร n3.3, n > 59ย , Kk Sc(n) + Z(n) <

3n n + = 2n, 2 2

dย nร รทv ย ยง. n < 59ย |^Oย ร ย ?1u รนยซย ยนe nร รทv ย ยง. nรพยครฃ, ยฝn3.2ยครก. eยกyยฒยฝn3.3. ยฉยฑenยซย ยน. (1) n = 2ฮฑ , ฮฑ > 1ย , Z(2ฮฑ ) = 2ฮฑ โ 1,

Sc(2ฮฑ ) > 1,

n = 2ฮฑ ร รทv ย ยง. (2) n = 2kpฮฑ , ฮฑ > 1, (2k, pฮฑ ) = 1, p > 3ย ย รชย , ร {ย ยง 4kx โ ก 1(modpฮฑ ) k),ย ร {ย ยง 16k 2 x2 โ ก 1(modpฮฑ ) k), ร )ร ย ย y,Kย 1 6 y 6 pฮฑ โ 1,qpฮฑ โ yยฝย cยกร {ย ยง ), Kย 1 6 y 6

(pฮฑ โ 1) .d 2

16k 2 y 2 โ ก 1(modpฮฑ ), Kpฮฑ |(4ky โ 1)(4ky + 1), (4ky โ 1, 4ky + 1) = 1,uยดpฮฑ |(4ky โ 1)ยฝpฮฑ |(4ky + 1).

32

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê Îą

ep |(4ky − 1), Kn =

2kpÎą 4ky(4ky−1) , 2

l k

4k(pÎą − 1) 1 Z(n) = m 6 4ky − 1 6 − 1 6 n − 2k − 1 6 1 − Îą n. 2 p

epÎą |(4ky + 1), Kn = 2kpÎą 4ky(4ky+1) , l k 2 4k(pÎą − 1) 1 Z(n) = m 6 4ky 6 6 n − 2k 6 1 − Îą n. 2 p

(3) n = (2k + 1)2Îą , Îą > 1, k > 1ž, KĂ“{Â?§ (2k + 1)x ≥ 1(mod2Îą+1 ) † (2k + 1)x ≥ −1(mod2Îą+1 ) Ăž7k), Â…)Â?Ûê, ÎąÂ?Ă“{Â?§ (2k + 1)x ≥ 1(mod2Îą+1 ) ), e1 6 Îą 6 2Îą − 1, K Îą=ÂŒ, Ă„K 2Îą + 1 6 Îą 6 2Îą+1 − 1, K 2Îą+1 − Îą 6 2Îą+1 − 2Îą − 1 = 2Îą − 1, Â…2Îą+1 − ιávĂ“{Â?§

(2k + 1)x ≥ −1(mod2Îą+1 ),

߇Ó{Â?§¼7k˜‡áv 1 6 Îą 6 2Îą − 1 )Îą, K2Îą+1 |[(2k + 1)Îą + 1]½2Îą+1 |[(2k + 1)Îą − 1].

e2Îą+1 |[(2k + 1)Îą + 1],K2Îą+1 (2k + 1)|[(2k + 1)Îą + 1](2k + 1), l 1 Îą Z(n) 6 Îą(2k + 1) 6 (2 − 1)(2k + 1) 6 1 − Îą n. 2 2Îą+1 |[(2k + 1)Îą − 1]ž, Ă“nÂ?kZ(n) 6 1 − 21Îą n. oƒ, ĂŠu(2),(3)ߍÂœš, =

n = 2ι pι1 1 pι2 2 ¡ ¡ ¡ pιk k (ι > 1, ιi > 1, k > 2)

33

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Â?Ă™IOƒŠ)ÂŞ, q Îł = min{2Îą , pÎą1 1 , pÎą2 2 , ¡ ¡ ¡ , pÎąk k }, K

l q Îł <

√ 3

n = 2Îą pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k > q 3Îł , n,K 2 1 1 Z(n) 6 n 1 − Îł < n 1 − √ = n − n3 , 3 q n

Ăš , n¿ŠÂŒÂž, dĂšn3.4, 7

2

Sc(n) + Z(n) < n + n 12 + n − n 3 < 2n, =TnĂ˜áv Â?§. nÞ¤ã, ½n3.3¤å.

3.3 'ušÂ–SmarandacheŸê9Ă™ÊóŸê Â?§ ĂŠ?Âż ĂŞnĂ?Âś –SmarandacheŸêZ(n)½Ă‚Â?áv

m P

kU n Ă˜ Â

k=1

ĂŞm,=Z(n) = min{m : m ∈ N + , n| m(m+1) }. dZ(n) ½Ă‚N´íĂ‘Z(1) = 2

1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 8, Z(10) = 4, Z(11) = 10, ¡ ¡ ¡ .NþÆÜïĂ„ ŸêZ(n) 5Â&#x;, ˜ ­Â‡ ( JĂšĂ&#x;ÂŽ[30,79,80] :

(a)ĂŠu?Âż ĂŞn,kZ(n) > 1; (b)ĂŠu?ÂżĂ›ÂƒĂŞpĂš ĂŞk, kZ(pk ) = pk − 1, Z(2k ) = 2k+1 − 1; (c)enÂ??ÂżĂœĂŞ, KZ(n) = max{Z(m) : m|n}.

P Šz[27]Ăš\ ŸêZ(n) ÊóŸêZ ∗(n), òĂ™½Ă‚Â?ávnU m k=1 k Ă˜ n o m(m+1)  ÂŒ ĂŞm, =Z ∗ (n) = max m : m ∈ N + , |n . ~ XZ ∗ (n) c A ‡ Š 2

Â?Z ∗ (1) = 1, Z ∗ (2) = 1, Z ∗ (3) = 2, Z ∗ (4) = 1, Z ∗ (5) = 1, Z ∗ (6) = 3, Z ∗ (7) = 1, Z ∗ (8) = 1, Z ∗ (9) = 2, ¡ ¡ ¡ . dŠz[27, 82], Z ∗ (n)äkXe5Â&#x;:

(d)ĂźÂ‡Ă›ÂƒĂŞp, qeávp = 2q − 1, KZ(pq) = p; ep = 2q + 1, KZ(pq) = p − 1; (e)en = 3s t(sÂ??Âż ĂŞ,tÂ?ĂœĂŞ), KZ ∗ (n) > 2; (f)ʤk ĂŞa, b, kZ ∗ (ab) > max{Z(a), Z(b)}. ĂœŠ+ĂŻĂ†ĂŻĂ„Â˜aÂ?šŸêZ(n)9Ă™ÊóŸêZ ∗ (n) Â?§ Z(n) + Z ∗ (n) = n

ÂŒ)5, ÂżJĂ‘eÂĄ Ă&#x;ÂŽ: Ă&#x; ÂŽ : (A)Â?§(3-4)Â?kk Â‡óê). Â?NÂ?k˜‡óê)n = 6; 34

(3-4)

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

(B) §(3-4) ¤kÛê)7 Û êp (p > 5). 'ud¯K, ©z[83]éÙ?1 ïÄ, y² ß (B), ¿(¡ß (A)E´úm ¯K. !ò? Ú0 éùü ß ?1 ïÄ, |^Ð {Ú|Ü {y¢ ù ü ß (5, e¡ (Ø: ½ n 3.4 §(3-4) k óê)n = 6; §(3-4) ¤kÛê) n = pk , p ê, k ?¿ ê. y :©ü« ¹?Ø. 1) n óê , ©±e4« ¹5y²: 1.1) n = 2k (k > 1) , kZ(2k ) = 2k+1 − 1, Z ∗ (2k ) = 1.l Z(2k ) + Z ∗ (2k ) 6=

2k , 2k Ø´ §(3-4) ).

1.2) n = 2p(p Û ê) ,ep = 3,KZ(6) = 3, Z ∗ (6) = 3,l Z(6) + Z ∗ (6) = 6, n = 6´ §(3-4) );ep = 5,KZ(10) = 4, Z ∗ (10) = 4,l Z(10) + Z ∗ (10) = 8 6= 10, n = 10Ø´ §(3-4) );ep > 5,KZ ∗ (2p) = 1, = Z(2p) = 2p − 1 , §(3-4)k). KI2p| 2p(2p−1) ,dd 2p − 1 óê, gñ. 2 U

1.3) n = 2k pα (k,α ê) ,e2k pα m 3/XA = 2B ± 1 'Xª, K px = 2 · 2y ± 1,

(3-5)

Ø U 32y = 2 · px ± 1.ùpx,y´©O uα, k ê. ¯¢þ, o¬ 3Ø é xÚy¦ (3-5)ª¤á. ÷vª(3-5)¤á xÚy©OP Xe?Ø¥ b,d a,c: 1.3.1)epx = 2 · 2y − 1, KZ ∗ (n) = max{px } = pb = 2 · 2a − 1.Z(n) = m = n − Z ∗ (n) = 2k pα − pb = 2k pα − 2 · 2a + 1, Kk

k α

(2 p − pb )(2k pα − 2 · 2a + 2) 2 p

2 k α

k = 1 , Ka = 1, pb = 3,u´8( n = 2p /ª; k > 1 , dª(3-6) 2k pα−b |(2k pα−b − 1)(2k−1 pα − 2a + 1) w,, ÃØb = α ´b < α,óêÑØ U Ø2 Ûê È, d §(3-4)Ã). 1.3.2)epx = 2 · 2y + 1,K Z ∗ (n) = max{px − 1} = pd − 1 = 2 · 2c . 35

(3-6)

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Z(n) = m = n − Z ∗ (n) = 2k pÎą − pd + 1 = 2k pÎą − 2 ¡ 2c , K

k Îą

(2 p − 2 ¡ 2c )(2k pÎą − pd + 2) 2 p

2 k Îą

(3-7)

k = cž, dÂŞ(3-7)k2pÎą |(pÎą −2)(2k pÎą −pd +2), óêĂ˜ÂŒU Ă˜Ă›ĂŞ,Â?§(3-4)dÂžĂƒ ); k > cž, k2k−c pÎą |(2k−c−1pÎą −1)(2k pÎą −pd +2), pÎą |(2k−c−1pÎą −1)(2k pÎą −pd +2)Ă˜

ÂŒU¤å, l džÂ?§(3-4)Ăƒ).

1.3.3) eĂ˜ 3'XÂŞ(3-5). KZ ∗ (n) = 1. Z(n) 6= n − 1. Ă„KĂ“Âœš1.2)ÂĽ ?

Ă˜Â˜ , Ûê 2 Ă˜ gĂą.

1.4) n = 2k uvž, Ù¼(2k , u) = (u, v) = (v, 2k ) = 1.Px,yŠOÂ?u,v Ă?f,zÂ? uk ĂŞ. n ˆ‡Ă?fƒme 3/XA = 2B Âą 1 'XÂŞ, Â?ÂŒUÂ?x = 2 ¡ 2z Âą 1½x = 2y ¡ 2z Âą 1, Ă˜ÂŒU 32z = 2x Âą 1½2z y = 2x Âą 1. w, 3þÊx,y,zá veÂĄ?Ă˜¼¤ 9 Ă˜½Â?§:

1.4.1)ex = 2 ¡ 2z − 1,KZ ∗ (n) = max{x} = x1 = 2 ¡ 2z1 − 1.Z(n) = m = n − Z ∗ (n) = 2k uv − x1 = 2k uv − 2 ¡ 2z1 + 1. Ă˜Â”Â˜Â„5, u = ex1 , Kk

k

(2 uv − x1 )(2k uv − 2 ¡ 2z1 + 2) 2 uv

2 k

(3-8)

k = 1ž, kz1 = 1, ÂŒ 2ev|(2ev − 1)(uv − 1), , v|(2ev − x1 )(uv − 1)Ă˜ÂŒU¤å,

Ñgù; k > 1ž, k

2k ev|(2k ev − 1)(2k−1 uv − 2z1 + 1), l óê Ă˜Ă›ĂŞ gĂą. 1.4.2)ex = 2 ¡ 2z + 1, KZ ∗ (n) = max{x − 1} = x2 − 1 = 2 ¡ 2z2 . Z(n) = m = n − Z ∗ (n) = 2k uv − x2 + 1 = 2k uv − 2 ¡ 2z2 , K

k

(2 uv − 2 ¡ 2z2 )(2k uv − 2 ¡ 2z2 + 1) 2 uv

, 2 k

(3-9)

k = z2 ž, k2uv|(uv−2)(2k uv−2k+1 +1), óêĂ˜ÂŒU Ă˜Ă›ĂŞ, Ă‘gĂą; k > z2 ž, k

2k−z2 +1 uv (2k uv − 2)(2k uv − x2 + 2), 36

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

u|(2k uv − 2)(2k uv − x2 + 2)Ă˜ÂŒU¤å, Ă‘gĂą.

1.4.3)ex = 2y ¡ 2z − 1,KZ ∗ (n) = max{x} = x3 = 2y3 ¡ 2z3 − 1. Z(n) = m = n − Z ∗ (n) = 2k uv − x3 = 2k uv − 2y3 ¡ 2z3 + 1.

Ă˜Â”Â˜Â„5, u = tx3 , Kk

k

(2 uv − x3 )(2k uv − 2y3 ¡ 2z3 + 2) 2k uv

, 2

(3-10)

k = 1ž, kz3 = 1, ÂŒ 2tv|(2tv − 1)(uv − 2y3 + 1), , v|(2tv − 1)(uv − 2y3 + 1)Ă˜ ÂŒU¤å, Ă‘gĂą; k > 1, k2k tv|(2k tv − 1)(2k−1 uv − 2z3 y3 + 1), óê Ă˜Ă› ĂŞ gĂą.

1.4.4)e= 2y ¡ 2z + 1, KZ ∗ (n) = max{x − 1} = x4 − 1 = 2y4 ¡ 2z4 . Z(n) = m = n − Z ∗ (n) = 2k uv − x4 + 1 = 2k uv − 2y4 ¡ 2z4 , Ă˜Â”Â˜Â„5, v = sy4 , Kk

k

(2 uv − 2y4 ¡ 2z4 )(2k uv − 2y4 ¡ 2z4 + 1) k 2 uv

, 2

(3-11)

k = z4 , k2uv|(uv − 2y4 )(2k uv − 2z4 +1 y4 + 1), Ăł ĂŞ Ă˜ ÂŒ U Ă˜ Ă› ĂŞ, Ă‘ g Ăą;

k > z4 ž, k2k−z4+1 su|(2k−z4 su − 2)(2k uv − x4 + 2), u|(2k−z4 su − 2)(2k uv − x4 + 2)Ă˜ ÂŒU¤å, Ă‘gĂą.

1.4.5)eĂ˜ 3¹Þ4ÂŤ'XÂŞ, KZ ∗ (n) = 1. Z(n) 6= n − 1,Ă„KÂŹ Ă‘n − 1Â?Ăł

ĂŞ gĂą.

nÞ¤ã, Â?§(3-4) óê)kÂ…=k˜‡Â?n = 6. 2) nÂ?Ûêž, Š¹e3ÂŤÂœš?Ă˜: 2.1)w,n = 1Ă˜´Â?§(3-4) ). 2.2) n = pk (pÂ?Ă›ÂƒĂŞ, kÂ? ĂŞ)ž, ep = 3,KZ(n) = n − 1, Z ∗ (n) = 2,l

Z(n) + Z ∗ (n) 6= n; ep > 5,KZ(n) = n − 1, Z ∗ (n) = 1. n = pk ´Â?§(3-4) ).

2.3) nškĂľÂ‡Ă˜Ă“ÂƒĂ?fž, n = uv,Ăšp(u, v) = 1. x,yŠOÂ?u,v Ă?f.

Ă˜Â”Â˜Â„5, u > v. Še AÂŤÂœš?Ă˜: 2.3.1)e 3'XÂŞx = 2y − 1. Z ∗ (n) = max{x} = x1 = 2y1 − 1. Pu = bx1. Z(n) = m = n − Z ∗ (n) = uv − x1 = uv − 2y1 + 1, Kk

(uv − x1 )(uv − 2y1 + 2) uv

, 2 37

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

ddŒ

(bv − 1)(uv − 2y1 + 2) bv

, 2

w,Ă˜¤å, Ă‘gĂą. -

(bv − 1)(uv − 2y1 + 2) v

, 2

2.3.2)e 3'XÂŞx = 2y + 1. Z ∗ (n) = max{x − 1} = x2 − 1 = 2y2 . Pn = dy2 . Z(n) = m = n − Z ∗ (n) = uv − x2 + 1 = uv − 2y2 ,

Kk

(uv − x2 + 2)(uv − 2y2 ) , uv

2

2du|(du − 2)(uv − x2 + 2), u|(du − 2)(uv − x2 + 2), w,Ă˜¤å, Ă‘gĂą.

2.3.3)eĂ˜ 3/Xx = 2y Âą 1 'XÂŞ, KZ ∗ (n) = 1. -Z(n) = m = n − 1, w

,n n(n−1) ¤å. eydž mĂ˜´ávZ(n)½Ă‚  ÂŠ. 2 dĂ“{Â?§uX ≥ 1(modv)k). ÂŒ u2 X 2 ≥ 1(modv)k). Ă™)Ă˜Â” Â?Y , K

Œ 1 6 Y 6 v − 1. Kv|(uY − 1)(uY + 1). v|(uY − 1)ž, k

uY (uY − 1) n = uv

, 2

Z(n) = m 6 uY − 1 6 u(v − 1) − 1 < uv − 1; v|(uY + 1)ž, k

uY (uY + 1) n = uv

, 2

Z(n) = m 6 uY 6 u(v − 1) < uv − 1. m = n − 1Ă˜´ávZ(n)½Ă‚  ÂŠ.

nĂœ¹ÞAÂŤÂœš, Ûên´Â?§(3-4) ) Â…= n = pk , Ù¼p > 5Â?ÂƒĂŞ, kÂ? ĂŞ. nÞ¤ã, ½n3.4¤å.

38

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

3.4 'ušSmarandachep‡Ÿê†–SmarandacheŸê Â?§ m P ĂŠ?Âż ĂŞn,–SmarandacheŸêZ(n)½Ă‚Â?ÂŚ n Ă˜ k  êm, k=1

n o

=Z(n) = min m : n m(m+1) .~X, TŸê cA‡ŠÂ?:Z(1) = 1, Z(2) = 3, Z(3) = 2

2, Z(4) = 7, Z(5) = 5, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 9, Z(l0) = 4, Z(11) =

l0, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 31, Z(17) = 16, Z(18) =

8, Z(l9) = 8, Z(20) = 15, ¡ ¡ ¡ .

' u Ăš ˜ Âź ĂŞ, N Ăľ Æ Ăś ĂŻ Ă„ § 5 Â&#x;, Âż ˜ ­ ‡ ( J, „ Š

z[85 − 89]. ~X, ĂœŠ+3Šz[87]ÂĽĂŻĂ„ Â?§Z(n) = S(n), Z(n) + 1 = S(n) ÂŒ) 5, ¿‰Ñ Â?§ Ăœ ĂŞ), 'uÚ߇Â?§, 3 Ă– 2.5!ÂĽkÂ?[ ĂŁ.

3Šz[75]ÂĽ, Ăš? Smarandachep‡ŸêSc(n), Sc(n)½Ă‚Â?ávy|n!Â…1 6 y 6 m ÂŒ ĂŞm,=Sc(n) = max{m : y|n!, 1 6 y 6 m, m + 1†n!}. ~ X,Sc(n) c A ‡ Š Â?:Sc(1) = 1, Sc(2) = 2, Sc(3) = 3, Sc(4) = 4, Sc(5) = 6, Sc(6) = 6, Sc(7) = 10, Sc(8) = 10, Sc(9) = 10, Sc(l0) = 10, Sc(11) = 12, Sc(12) = 12, Sc(13) = 16, Sc(14) = 16, Sc(15) = 16, ¡ ¡ ¡ .

Š z[75]ĂŻ Ă„ Sc(n) Ă? 5 Â&#x;, Âż y ² Âą e ( Ă˜: eSc(n) = x, Â…n 6= 3,

Kx + 1´ÂŒun  ÂƒĂŞ.

m P k Ă˜n 3Šz[91]ÂĽĂš? –SmarandacheÊóŸêZ ∗ (n), Z ∗ (n) ½Ă‚Â?áv k=1 n o  ÂŒ ĂŞm, =Z ∗ (n) = max m : m(m+1) |n . Šz[272]ĂŻĂ„ Z ∗ (n) 5Â&#x;, 2

˜ ­Â‡ (J. Šz[86]ÂĽĂŻĂ„ Ăšn‡Ÿêƒm 'XÂ?§Z(n) + Z ∗ (n) = n („3.3!)†Sc(n) = Z ∗ (n) + n, ˜ ­Â‡(J, ÂżJĂ‘ ˜ „™)Ăť Ă&#x;ÂŽ: Ă&#x; ÂŽ :Â?§Sc(n) = Z ∗ (n) + n )Â?pÎą , Ù¼pÂ?ÂƒĂŞ,2†ι, pÎą + 2Â?Â?ÂƒĂŞ. ! 8 ´ïĂ„¹Þ¯K, eÂĄ : ½ n 3.5 Â?§Sc(n) = Z ∗ (n) + n )Â?pÎą , Ù¼pÂ?ÂƒĂŞ, 2†ι, pÎą + 2Â?Â?ÂƒĂŞ, Âą9á v^‡ι(2Îą − 1)†n(Îą > 1), n + 2 Â?ÂƒĂŞ, nÂ? ĂŞ. 3y²½nƒck‰ÑeÂĄ A‡Ún. Ăš n 3.5 eSc(n) = x ∈ Z,Â…n 6= 3,Kx + 1Â?ÂŒun  ÂƒĂŞ. y ² :„Šz[75]. ddŒ„, Sc(n)Ă˜ 3n = 1, n = 3Â?Ûê , 3Ă™{Âœše ŠÑ´óê. Ăš n 3.6 Z ∗ (pÎą ) =

  2, p 6= 3  1, p = 3

39

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ăš n 3.7 en ≥ 0(modÎą(2Îą − 1)),KkZ ∗ (n) > 2Îą > 1. Ăš n 3.8 √

8n + 1 − 1 . 2 Ăšn3.6-Ăšn3.8 y²ÂžĂŤ Šz[27]. ∗

Z (n) 6

Ăš n 3.9 n = pÎą0 0 pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k (p0 = 2, pi > 3, k > 1, Îąi > 1)Â?n IOƒŠ)ªž,k n

Z(n) 6 n −

min{pι0 0 , pι1 1 , pι2 2 , ¡ ¡ ¡ , pιk k } y ² :aqu3.3!¼y²½n3.3 �{. n = pι0 0 pι1 1 pι2 2 ¡ ¡ ¡ pιk k (p0

= 2, pi > 3, k >

1, Îąi > 1)Â?Ă™IOƒŠ)ªž, ŠßÂœš5y².

(i) n = 2kpÎą , Îą > 1, (2k, pÎą ) = 1, p > 3Â?ÂƒĂŞ, dĂ“{Â?§ 4kx ≥ 1(modpÎą ) k), ÂŒ Ă“{Â?§ 16k 2 x2 ≥ 1(modpÎą ) k ), Ă™ ) Ă˜ ” Â?y,K ÂŒ 1 6 y 6 pÎąâˆ’1 ,qpÎąâˆ’y ½ Â? c ÂĄ Ă“ { Â? § ), K ÂŒ 1 6 y 6

pÎąâˆ’1 .d 2

16k 2 y 2 ≥ 1(modpÎą ), KpÎą |(4ky − 1)(4ky + 1), (4ky − 1, 4ky + 1) = 1,u´pÎą |4ky − 1 ½pÎą |4ky + 1 .

Îą Îą 4ky(4ky−1) ep |4ky − 1 , Kn = 2kp

, l 2

Z(n) = m 6 4ky − 1 1 4k(pÎą − 1) 6 − 1 6 n − 2k − 1 6 1 − Îą n 2 p n 6n − . min{pÎą0 0 , pÎą1 1 , pÎą2 2 , ¡ ¡ ¡ , pÎąk k }

epÎą |4ky + 1 , Kn = 2kpÎą 4ky(4ky+1) , l Â?k 2 4k(pÎą − 1) 1 n Z(n) = m 6 4ky 6 6 n − 2k = 1 − Îą n 6 n − . Îą0 Îą1 Îą2 2 p min{p0 , p1 , p2 , ¡ ¡ ¡ , pÎąk k } (ii) n = 2Îą (2k + 1), (Îą > 1, k > 1),KĂ“{Â?§ (2k + 1)x ≥ 1(mod2Îą+1 ) † (2k + 1) ≥ −1(mod2Îą+1 ) 40

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

Ăžk), Â…)Â?Ûê, ÎąÂ?Ă“{Â?§ (2k + 1)x ≥ 1(mod2Îą+1 ) ), e1 6 Îą 6 2Îą − 1,K Îą=ÂŒ, Ă„K 2Îą + 1 6 Îą 6 2Îą+1 − 1, K 2Îą+1 − Îą 6 2Îą+1 − 2Îą − 1 = 2Îą − 1, Â…2Îą+1 − ιávĂ“{Â?§

(2k + 1)x ≥ −1(mod2Îą+1 ),

߇Ó{Â?§¼7k˜‡áv1 6 Îą 6 2Îą −1 )Îą,K2Îą+1 |(2k + 1)Îą + 1½2Îą+1 |(2k + 1)Îą − 1 , e2Îą+1 |(2k + 1)Îą + 1 ,K

2Îą+1(2k + 1) |[(2k + 1)Îą + 1](2k + 1)Îą, l 1 n Z(n) 6 Îą(2k + 1) 6 (2 − 1)(2k + 1) 6 1 − Îą n 6 n − . Îą0 Îą1 Îą2 2 min{p0 , p1 , p2 , ¡ ¡ ¡ , pÎąk k } Îą

2Îą+1 |(2k + 1)Îą − 1ž, Ă“nÂ?k 1 n . Z(n) 6 1 − Îą n 6 n − Îą0 Îą1 Îą2 2 min{p0 , p1 , p2 , ¡ ¡ ¡ , pÎąk k }

nĂœ(i),(ii)k, n = pÎą0 0 pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k (p0 = 2, pi > 3, k > 1, Îąi > 1)Â?Ă™IOƒŠ)

ªž, K

Z(n) 6 n −

n min{pι0 0 , pι1 1 , pι2 2 , ¡ ¡ ¡

, pÎąk k }

.

e¥ò‰Ñ½n y², ŠĂŠÂŤÂœš5y². (1)n = 1ž, Z ∗ (1) = 1, Sc(1) = 1,K1Ă˜Â?Ă™). (2)n = 3Îą (Îą > 1),d Ăš n3.6, Z ∗ (3Îą ) = 2,en = 3Îą ´ Â? § ),KSc(3Îą ) = 2 + 3Îą ,Ă?Â?3|3Îą + 2 + 1,l 3Îą + 2 + 1Ă˜ÂŒUÂ?ÂƒĂŞ †Ún3.5ƒgĂą, n = 3Îą Ă˜´ Â?§ ). (3)n = pÎą (Îą > 1, p > 5),dĂšn3.6,Z ∗ (pÎą ) = 1, en = pÎą ´ Â?§ ), KSc(pÎą ) = β

1 + pι ,� p > 5ž, 3|p2

+2

, dĂšn3.5, ÎąĂ˜UÂ?óê, Â… pÎą + 2(2†ι) Â?ÂƒĂŞÂž,

n = pÎą (Îą > 1, p > 5Â?ÂƒĂŞ)áv Â?§. (4)n = 2Îą (Îą > 1),e m(m+1) |2Îą ,Ă?(m, m + 1) = 1,Km = 1, Z ∗ (2Îą ) = 1,en = 2Îą ´ 2

Â?§ ),KSc(2Îą ) = 1 + 2Îą ,Ă?2|(2Îą + 1 + 1),†Ún3.5gĂą, n = 2Îą (Îą > 1)Ă˜´ 41

Smarandache ¼ê9Ù '¯KïÄ

§ ). (5)n = pα1 1 pα2 2 · · · pαk k (k > 2, αi > 1) ÙIO ©)ª. q© ü« ¹5y².

(i)2†n,K2†pαi i ,l 2|Sc(n),en ÷v §,K7L2†Z ∗ (n).y ÄZ ∗ (n),e

3 êα(α > 1),¦α(2α − 1)|n,KZ ∗ (n) > 2α − 1,e 3 êα(α > 1),¦α(2α + 1)|n, KZ ∗ (n) > 2α,l

Z ∗ (n) = max{max{2k : k(2k + 1)|n}, max{2k − 1 : k(2k − 1)|n}}. 2©n« ¹5?Ø 1 ,eZ ∗ (n) = 2α − 1 > 1,Kα(2α − 1)|n,kα|n, α|[n + (2α − 1) + 1]. en ÷v

§, KSc(n) = 2α − 1 + n. Sc(n) + 1Ø ê, Ún3.5 gñ.

1 , eZ ∗ (n) = 2α > 1,Kα(2α + 1)|n,kα|n, (2α + 1)|n.en ÷ v §,

KSc(n) = 2α + n. Sc(n) + 1Ø ê, Ún3.5 gñ.

, eZ ∗ (n) = 1, dα > 1, Kα(2α − 1)†n, l en + 2Ø´ ê,dÚn3.5, ù

n Ø´ § ). en + 2 ê, dÚn3.5, ù n § ). =α(2α − 1)†n, n + 2 ê ên § ).

(ii)2|n,en÷v §, K7LZ ∗ (n) óê, Z ∗ (n) > 2, Z ∗ (n) = m > 2,

m(m + 1) |n , 2

K(m + 1)|n,? (m + 1)|(n + m + 1), ù Sc(n) = n + m + 1Ø´ ê, Ún3.5gñ. Ïd, §Sc(n) = Z ∗ (n) + n ) pα , Ù¥p ê, 2†α, pα + 2 ê, ±9÷v^ α(2α − 1)†n(α > 1), n + 2 ê, n ê. ùÒ ¤ ½n3.5 y².

3.5 'uSmarandacheV ¦¼ê é ? ¿ ên,Í ¶ Smarandache¼ êZ(n)½  êm¦ n o m(m+1) m(m+1) n| 2 , =Z(n) = min m : m ∈ N, n| 2 .ù ¼ê´dÛêZæͶêØ

;[Smarandache3©z[7]¥Ú? . 'u¼êZ(n) ê5 , NõÆö?1 ïÄ, ¼ Ø k (J[28,36,97−99] . ~X: (1)éu?¿ ên,Z(n) < nØð¤á. (2)é?¿ êp > 3, Z(p) = p − 1.

(3)é?¿ êp > 39k ∈ N, Z(pk ) = pk − 1. p = 2 , KkZ(2k ) = 2k+1 − 1.

(4)Z(n)´Ø \ , =Z(m + n)Øð uZ(m) + Z(n);Z(n) Ø´ ¦ , =Z(m ·

n)Øð uZ(m) · Z(n).

l±þ {ü 5 ±wÑ, Z(n) ©ÙéØ5Æ, 'u§ 5 k u

42

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

? ÚïÄ. d Smarandache ½  , ê Ø ¼ êSdf (n) Sdf (n) = min{m : m ∈

N, n|m!!}. T¼ê¡ SmarandacheV ¦¼ê, 'u§ Ð 5 , kØ Æö ?1 ïÄ¿ ­ (J[4,6,31,100] . ~X, Sdf (n)k é­ 5 ,

=Sdf (n) ±Ûó5ØC, Ò´`en Ûê, KSdf (n) Ûê, en óêKSdf (n) óê. éN´uy, Sdf (n) ´ éØ5Æ ¼ê, cÙ´3n ó ê ÿSdf (n)LyÑéØ­½ 5 . éZ(n)ÚSdf (n), ùü þLyÑØ­½5 ¼ê, UÄ3§ méÑ é XQ? ! Ì 8 Ò´ïÄ e¡ü ¼ê § )5, =¦ § Z(n) = Sdf (n), Z(n) + 1 = Sdf (n) ¤k ê). !ÏLÐ {0 ù ¯K ) , =y² e¡ ü ½n. ½ n 3.6 é?¿ ên > 1, ¼ê § Z(n) = Sdf (n)

(3-12)

=kÛê), Ù¤k)=k2«/ª: n = 45; n = pα1 1 pα2 2 · · · pαk k p,

Ù¥k > 1, p1 < p2 < · · · < pk < pþ Û ê, é1 6 i 6 k, αi > 1,Ó p + 1 ≡ 0(modpαi i ).

½ n 3.7 é?¿ ên > 1, ¼ê § Z(n) + 1 = Sdf (n)

(3-13)

=kÛê), Ù¤k)=k3«/ª: n = 9; n = p; n = pα1 1 pα2 2 · · · pαk k p,

Ù¥k > 1, p1 < p2 < · · · < pk < p þ Û ê, é1 6 i 6 k, αi > 0 3 αi > 1, Ó p − 1 ≡ 0(modpαi i ).

e¡y²ùü ½n. Äk½n3.6. ¯¢þ, n = 1 , §Z(n) = Sdf (n)¤á. éN´ y, n = 2, 3, 4, 5, 6, 7 , Z(n) = Sdf (n)ؤá. u´Ø n > 8. Äky² §(3-12)Ø Ukóê). n = 2α pα1 1 pα2 1 · · · pαk k n IO©)Ϫ.

(1)ek = 0,=n = 2α , d dSdf (n) 5 Sdf (2α ) óê, Z(2α ) = 2α+1 −

1 Ûê, l Sdf (2α ) 6= Z(2α ). d nØ÷v §. 43

Smarandache ¼ê9Ù '¯KïÄ

(2)ek > 1,Kn = 2α pα1 1 pα2 1 · · · pαk k .dSdf (n) ± Û ó 5 Ø C , eZ(n) =

Sdf (n) = a,Ka7 óê, Ø Z(n) = Sdf (n) = 2m,u´dZ(n)½Âk 2α pα1 1 pα2 1 · · · pαk k |m(2m + 1).

(3-14)

αi α α d(3-14)ª , 7k2α |m, é∀pαi i 7kpi |m ½pi |(2m + 1). d2 |m íÑ

2α |m!!. α

(3-15)

α

PS1 = {pαi i : pαi i |m}, S2 = {pj j : pj j |(2m + 1)}.

5¿ ¯¢, =S2 6= Ø,ù´Ï eS2 = Ø,Ké¤kpαi i þkpαi i |m, q2α |m, u

´d2α pα1 1 pα2 2 · · · pαk k | (2m−1)·2m , Z(n) 6 2m − 1, S2 6= Øu´: 2

(i)eS1 = Ø,=∀pαi i , pαi i |(2m + 1). emax{pαi i } 6 m,(Ü(3-15)ª , Sdf (n) 6 m; αi

αi

emax{pαi i } > m,(Ü(3-15)ª , Sdf (n) = 2s, 7∃pi0 0 ∈ S2 ,¦ pi0 0 |s. ,m 6= s.l Sdf (n) 6= 2m;

α

α

(ii)eS1 6= Ø, pαi i ∈ S1 , pj j ∈ S2 . emax{pαi i } < max{pj j },aq(i) ©Û α

α

{ , ÃØmax{pj j } 6 m, ´max{pj j } > m,ÑkSdf (n) 6= 2m. emax{pαi i } > α

max{pj j } , íÑSdf (n) 6 m.

nÜ(i),(ii) , d Sd(n) 6= Z(n).ùÒy² §(3-12)Ø Ukóê).

en Ûê, n = pα1 1 pα2 1 · · · pαk k pα n IO©)Ϫ. ±e©3« ¹?Ø:

(1)ek = 0,=n = pα ,d Z(pα ) = pα − 1 ó, Sdf (n) Û, ¤ ±Z(pα ) 6=

Sdf (pα ).

(2)ek > 1 α = 1,Kn = pα1 1 pα2 1 · · · pαk k p.

emax{pαi i } < p, KdSdf (n)5

Sdf (n) = Sdf (pα1 1 pα2 1 · · · pαk k p) = max{Sdf (pα1 1 ), · · · , Sdf (pαk k ), Sdf (p)} = p. e¤áZ(n) = Sdf (n) = p, KdZ(n)½Â pα1 1 pα2 1

p(p · · · pαk k p

+ 1) . 2

u´7L¤ápαi i |(p + 1), =p + 1 ≡ 0(modpαi i ).

, ¡, y² ÷vp + 1 ≡ 0(modpαi i ), ÒkZ(n) = p.é∀h < p, (i)eh < p − 1,Kdp †

h(h+1) , 2

h(h+1) , Z(n) 6= h; 2 2 , pα1 1 pα2 1 · · · pαk k p† (p−1)p 2 ,

pα1 1 pα2 1 · · · pαk k p †

(ii)eh = p−1,Kdpαi i |(p+1)9(p−1, p+1) =

Z(n) 6=

p − 1.l , ep + 1 ≡ 0(modpαi i ), KkZ(n) = Sdf (n), ù=´ §(3-12) |). α

emax{pαi i } > p, Kd Sdf (n) = p, ½Sdf (n) = max {Sdf (pαi i )} = Sdf (pj j ). (i)eSdf (n) = p,Kdmax{pαi i } > p , max{pαi i } † 44

16i6k p(p+1) , 2

Z(n) 6= p,l Z(n) 6=

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

Sdf (n). α

α

(ii)eSdf (n) = max {Sdf (pαi i )} = Sdf (pj j ), dSdf (n) 5 ,7kpj |Sdf (pj j ), 16i6k

u´pj |Sdf (n).

Äk5¿ ¯¢, =é?¿Û êp9α > 2, Øpα = 32 , þkSdf (pα ) < pα ,

Ï ep 6 2α − 1, KSdf (pα ) = (2α − 1)p < pα (Øpα = 32 ); ep < 2α − 1, KéN´ dp2 |pα !! , Sdf (pα ) < pα . Sdf (n) = max {Sdf (pαi i )} = Sdf (32 ) = 9 , éN´ 16i6k

2

yn = 3 × 5 = 45 ), Ù{ ¹þؤá.

α

α

é uSdf (n) = max {Sdf (pαi i )} = Sdf (pj j ) < pj j , e ¤ áZ(n) = Sdf (n) =

m,Km < α

16i6k αj α1 α1 pj p1 p2 · · · pαk k p| m(m+1) . 2

pj j gñ. Z(n) 6= Sdf (n).

α

q d ± þ © Û , pj |m,l pj j |m,ù m <

(3)ek > 1 α > 2, Kn = pα1 1 pα2 1 · · · pαk k pα .

eSdf (n) = max {Sdf (pαi i )} = Sdf (pα ), dk > 1 pα 6= 32 ,u ´Sdf (n) = 16i6k

α

α

Sdf (p ) = m < p .d dp|m9m < pα íÑpα †

Sdf (n).

m(m+1) , 2

ÏdZ(n) 6= m, Z(n) 6=

α

eSdf (n) = max {Sdf (pαi i )} = Sdf (pj j ), dα > 2 , 3 ù « ¹ eαj > 16i6k

α 2 pj j

α

2

α

α

6= 3 ,u ´Sdf (n) = Sdf (pj j ) = m < pj j . Ó , dpj j †

m, Z(n) 6= Sdf (n).

m(m+1) , 2

Z(n) 6=

u´B ¤ ½n3.6 y². e¡y²½n3.7. ½n3.6 y² { q, ùü Ñ VL§. dZ(p) = p −

19Sdf (p) = p ,n = p÷v §(3-13). én = pk /, Z(pk ) + 1 = pk − 1 + 1 = pk , ´Øn = 32 þ¤áSdf (pk ) < pk , ,Ø U÷v §(3-13). n = 32 = 9TÐÒ

´ §(3-13) AÏ). én = pα1 1 pα2 1 · · · pαk k p /, N´y², ¦Z(n) + 1 =

Sdf (n) = ÷vp − 1 ≡ 0(modpαi i ). Ù{ ¹þ ½n3.6 y² {aq,ùpò Ø2Kã.

3.6 a2 Smarandache¼ê Ͷ Smarandache ¼êZ(n)½Â :

m(m + 1) }. Z(n) = min{m : m ∈ N + , n

2

(3-16)

© z[105]í 2 Smarandache¼ ê, é ? ¿ ên,í 2 Smarandache¼ êZ3 (n)½Â

m(m + 1)(m + 2) Z3 (n) = min{m : m ∈ N , n

}. 6 +

45

(3-17)

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

! ò 0 ' uZ3 (n) Ă? 5 Â&#x; ĂŻ Ă„, Š O ĂŠn = 2l p, n = 2l pk , n = 2l Ă—

3k , l, k ∈ N + , p > 3Â?ÂƒĂŞ, ‰Ñ Z3 (n) ŸêŠ/ÂŞ. ÚʟêÂ?§ ĂŞ), ފ9 ĂŹCúª ÂŻK ĂŻĂ„Â?´k¿Â [29,36,107] .

Ăš n 3.10 ĂŠ?Âż ĂŞk, ÂƒĂŞp, eZ3 (kp) = m, Km7Â?lp − 2, lp − 1, lp, l ∈ N + nÂŤ Âœ/.

y ² :Ă?Â?Z3 (n) > 1, p = 2, 3ž, Ăšnw,¤å. p > 3´ÂƒĂŞ, dp|m(m + 1)(m + 2)/6, Kp7 Ă˜m, m + 1, m + 2Ă™ÂĽÂƒÂ˜, Ă?dm7klp − 2, lp − 1, lp, l ∈ N + nÂŤÂœ/. y..

e¥ò‰Ñ'uZ3 (n) A‡{Ăź5Â&#x;9y². (1) n = 2l p, l ∈ N + , p > 3Â?ÂƒĂŞ, Z3 (2l p) = m, Ă„m = gp−2, m = gp−1, m =

gpnÂŤÂœš9g Šk:

(a)eg = 1, m = p − 2,dž=km + 1 = p − 1Â?óê, d2l p|(p − 2)(p − 1)p/6,K

k2l+1 |p − 1,^ Ă“ { ÂŞ L ĂŁ = Â?, p − 1 ≥ 0(mod2l+1 )ž,Z3 (2l p) = p − 2;em =

p − 1, d2l p|(p − 1)p(p + 1)/6,Kk2l |p − 1½Ü2l |p + 1,= p Âą 1 ≥ 0(mod2l ), p − 1 6≥

0(mod(2l+1 ))ž, Z3 (2l p) = p − 1;e2l p|p(p + 1)(p + 2)/6,Kkp + 1 ≥ 0(mod(l + 1)), d žp + 1 ≥ 0(mod2l ),AkZ3 (2l p) = p − 1.

(b)Ă“n, eg = 2, m = 2p − 2ž, d2l p|(2p − 2)(2p − 1)2p/6,K2l−1 |p − 1,= p − 1 ≥

0(mod2),Â…p − 1 6≥ 0(mod2l )ž, Z3 (2l p) = 2p − 2;m 6= 2p − 1,Ú´Ă?Â?em = 2p −

1,K2l p|(2p−1)2p(2p+1)/6,Ú´Ă˜ÂŒU ;em = 2p,d2l p|2p(2p+1)(2p+2)/6,K2l−1 |p+ 1,=p + 1 ≥ 0(mod2l−1 ),Â…p + 1 6≥ 0(mod2l ).

(c)eg = 2k , k > 1, k ∈ N,d2l p|(2k p − 2)(2k p − 1)2k p/6, K2l |2k ,dZ3 (n)  5,

2k = 2l , dž, Z3 (2l p) = 2l p − 2.

(d)eg > 3Â?Ûê, m = gp − 2,d2l p|(gp − 2)(gp − 1)gp/6, k2l+1 |gp − 1;em =

gp − 1,d2l p|(gp − 1)gp(gp + 1)/6,K2l−1 |gp − 1½Ü2l |gp + 1;em = gp,d2l p|gp(gp +

1)(gp + 2)/6, K2l+1 |gp + 1,½k2l |gp + 1, džAkm = gp − 1.Ă“n, eg > 3Â?Ûê, m = 2gp − 2,džk2l−1 |gp − 1;em = 2gp, g > 3Â?Ûê, džk2l−1 |gp + 1.–dmÂŽv

46

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

kÙ§ U /. nþ, k   p − 2, p − 1 ≡ 0(mod2l+1 );       p − 1, p ± 1 ≡ 0(mod2l );      2p − 2, p − 1 ≡ 0(mod2l−1 );       p + 1 ≡ 0(mod2l−1 );   2p, Z3 (2l p) = gp − 2, gp − 1 ≡ 0(mod2l+1 );     gp − 1, gp ± 1 ≡ 0(mod2l );       2sp − 2, sp − 1 ≡ 0(mod2l−1 );       2sp, sp + 1 ≡ 0(mod2l−1 );     l 2 p − 2, Ù§,

Ù¥3 6 g 6 2l − 1, 3 6 s 6 2l−1 − 1 Ûê.

(2)Ón, n = 2l pk , l ∈ N + , k ∈ N + , p > 3 ê , k   pk − 2, pk − 1 ≡ 0(mod2l+1 );       pk − 1, pk ± 1 ≡ 0(mod2l );      2pk − 2, pk − 1 ≡ 0(mod2l−1 );      k  pk + 1 ≡ 0(mod2l−1 );   2p , Z3 (2l pk ) = gpk − 2, gpk − 1 ≡ 0(mod2l+1 );     gpk − 1, gpk ± 1 ≡ 0(mod2l );       2spk − 2, spk − 1 ≡ 0(mod2l−1 );       2spk , spk + 1 ≡ 0(mod2l−1 );     l k 2 p − 2, Ù§,

Ù¥3 6 g 6 2l − 1, 3 6 s 6 2l−1 − 1 Ûê.

n = 2l × 3k , l, k ∈ N + , Ä þª9©z[105]¥5 2 (©z[105]¥[4]ª)k   3k+1 − 2, 3k+1 − 1 ≡ 0(mod2l+1 );       3k+1 − 1, 3k+1 ± 1 ≡ 0(mod2l );      2 × 3k+1 − 2, 3k+1 − 1 ≡ 0(mod2l−1 );      k+1  3k+1 + 1 ≡ 0(mod2l−1 ); 2 × 3 ,  Z3 (2l × 3k ) = g · 3k+1 − 2, g · 3k+1 − 1 ≡ 0(mod2l+1 );     g · 3k+1 − 1, g · 3k+1 ± 1 ≡ 0(mod2l );       2s · 3k+1 − 2, s · 3k+1 − 1 ≡ 0(mod2l−1 );       2s · 3k+1 , s · 3k+1 + 1 ≡ 0(mod2l−1 );     l 2 × 3k+1 − 2, Ù§, 47

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ù¼3 6 g 6 2l − 1, 3 6 s 6 2l−1 − 1Â?Ûê.

3.7 ˜‡Â?šSmarandacheŸê91 a–SmarandacheŸê Â?§ ĂŠ?Âż ĂŞn, Ă?Âś SmarandacheŸêÂ?S(n)½Ă‚Â? êmÂŚ n|m!. =S(n) = min{m : m ∈ N, n|m!}. 1 a–SmarandacheŸêZ2 (n)½Ă‚Â? êkÂŚ n Ă˜ k

2

(k+1)2 ,½Ü 4

2

k (k + 1)2 , Z2 (n) = min k : k ∈ N, n

4

Ù¼NL¤k ĂŞ 8Ăœ. Ú߇Ÿê¹9k'SmarandacheŸê ½Ă‚ÂŒĂŤ Š z[7]. lS(n)9Z2 (n) ½Ă‚N´íĂ‘§Â‚ cA‘ŠÂ?: S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, ¡ ¡ ¡ . Z2 (1) = 1, Z2 (2) = 3, Z2 (3) = 2, Z2 (4) = 3, Z2 (5) = 4, Z2 (6) = 3, Z2 (7) = 6, Z2 (8) = 7, ¡ ¡ ¡ . 'uS(n) Ă? 5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âź Ă˜ k¿Â (J[28,36,97−100] . ~ XŠz[28]ĂŻĂ„ Â?§ S(m1 + m2 + ¡ ¡ ¡ + mk ) =

k X

S(mi )

i=1

ÂŒ)5, |^)Ă›ĂŞĂ˜ÂĽĂ?Âś nƒê½ny² ĂŠ?Âż ĂŞk > 3, TÂ?§kĂƒÂĄ Ăľ| ĂŞ)(m1 , m2 , ¡ ¡ ¡ , mk ).

Šz[36]ĂŻĂ„ S(n) ŠŠĂ™ÂŻK, y² ĂŹCúª 3

X

n6x

2Îś( 32 )x 2 (S(n) − P (n))2 = +O 3 ln x

3

x2 ln2 x

!

,

Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f,Îś(s)LÂŤRiemann zeta-Ÿê. Šz[97 − 98]ĂŻĂ„ S(2p−1 (2p − 1)) e. OÂŻK, y² ĂŠ?ÂżÂƒĂŞp > 7, k

OÂŞ

S(2p−1 (2p − 1)) > 6p + 19S(2p + 1) > 6p + 1.  C, Šz[99]Âź Â?˜„ (Ă˜:=y² ĂŠ?ÂżÂƒĂŞp > 17Ăš?ÂżĂ˜Ă“ ĂŞa 9b,k OÂŞ S(ap + bp ) > 8p + 1. d , Šz[100]?Ă˜ SmarandacheŸê ,Â˜ÂŤe. OÂŻK, =SmarandacheÂź

48

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

êʤ êê e. O¯K, y² Ê?¿ ên > 3k Oª: n

S(Fn ) = S(22 + 1) > 8 ¡ 2n + 1, n

Ù¼Fn = 22 + 1Â?Ă?Âś ¤ ĂŞĂŞ. 'uS(n) Ă™§ïĂ„SNš~ÂƒĂľ, ĂšpĂ˜2˜˜ Ăž. ĂŠuŸêZ2 (n) 5Â&#x;, –8 ) ĂŠ , $Â–Ă˜ ڇŸê ފ´Ă„äkĂŹC5Â&#x;. ! ĂŒÂ‡8 ´|^Ă? Â?{ĂŻĂ„ŸêÂ?§Z2(n) + 1 = S(n) ÂŒ)5, ¿Ÿ

ڇÂ?§ ¤k ĂŞ), äN/`Â?Ă’´y² eÂĄ : ½ n 3.8 ĂŠ?Âż ĂŞn,ŸêÂ?§Z2 (n) + 1 = S(n)kÂ…=ke nÂŤ/ÂŞ ): (A)n = 3, 4, 12, 33 , 2 ¡ 33 , 22 ¡ 33 , 23 ¡ 33 , 24 ¡ 33 , 34 , 2 ¡ 34 , 22 ¡ 34 , 23 ¡ 34 , 24 ¡ 34 ; 2

(B)n = p ¡ m,Ù¼p > 5Â?ÂƒĂŞ, mÂ? Ă˜ (p−1) ?Âż ĂŞ; 4

(C)n = p2 ¡ m,Ù¼p > 5Â?ÂƒĂŞÂ…2p − 1Â?ĂœĂŞ, mÂ?(2p − 1)2 ?ÂżÂŒu1 Ă?ĂŞ. w,T½n”.)Ăť Â?§Z2(n) + 1 = S(n) ÂŒ)5ÂŻK. ½=y² ڇÂ?§ kĂƒÂĄĂľÂ‡ ĂŞ)¿‰Ñ § z‡) äN/ÂŞ. eÂĄ|^Ă? Â?{Âą9SmarandacheŸê 5Â&#x;‰Ñ½n † y². k'g,ĂŞ Ă˜5Â&#x;Âą9ÂƒĂŞ k'SNÂŒĂŤ Šz[4 − 6]. ¯¢ÞN´ y: n = 3, 4, 12, 33 , 2 ¡ 33 , 22 ¡ 33 , 23 ¡ 33 , 24 ¡ 33 , 34 , 2 ¡ 34 , 22 ¡ 34 , 23 ¡ 34 , 24 ¡ 34 ž, Â?§Z2 (n) + 1 = S(n)w,¤å. y3ŠeÂĄAÂŤÂœšÂ?[?Ă˜: S(n) = S(p) = p > 5ž, n = mp,KS(m) < pÂ…(m, p) = 1.dženávÂ? §Z2 (n)+1 = S(n),@oZ2 (n) = p−1,¤¹dZ2 (n) ½Ă‚kn Ă˜ (p−1) 4 2 (p−1)2 Âąm| (p−1) 4 ,Ă? dmÂ? 4 ? Âż Ă? ĂŞ.

‡ ƒ, n =

2 2

p

,=mp| (p−1) 4

2 mp…m| (p−1) 4 ž,

2 2

p

.¤

kS(n) =

p, Z2 (n) = p − 1, ¤¹n = mp´Â?§Z2 (n) + 1 = S(n) ). u´y² ½nÂĽ 1 ÂŤÂœš(B).

S(n) = S(p2 ) = 2p, p > 5ž, n = mp2 ,KS(n) < 2pÂ…(m, p) = 1.dženáv Â?§Z2 (n) + 1 = S(n),@oZ2 (n) = 2p − 1,¤¹dZ2 (n) ½Ă‚kn Ă˜(2p − 1)2 p2 ,¤¹ mp2 |(2p − 1)p2 , ½Üm|(2p − 1)2 . w ,m 6= 1, Ă„ Kn = p2 , S(p2 ) = 2p, Z2 (p2 ) = p − 1.¤ Âą d žn = p2 Ă˜ á v Â? §Z2 (n) + 1 = S(n).u´m7L´(2p − 1)2 ˜‡Œu1 Ă?ĂŞ. d , Ă?Â?2p − 1Â?Ăœ ĂŞ, n = mp2 ž, Z2 (n) 6= p − 1, Z2 (n) 6= p, ¤¹Z2 (n) = 2p − 1, S(n) = 2p, ¤ Âąn = mp2 ávÂ?§Z2 (n) + 1 = S(n).u´y² ½nÂĽ Âœ/(C).

y3y² S(n) = S(pÎą )Â…p > 5Âą9Îą > 3ž, nĂ˜ávÂ?§Z2 (n) + 1 = S(n). Ăš ž n = mpÎą , S(m) 6 S(pÎą ), (m, p) = 1. u´kS(n) = hp, Ăšph 6 Îą.enávÂ? 49

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

§Z2 (n) + 1 = S(n),KZ2 (n) = hp − 1. u´dZ2 (n) ½Ă‚k

2 2 2

Îą (hp − 1) m p n = mp

4

l d Ă˜ 5Â&#x;ÂŒ pÎąâˆ’2 |h2 6 Îą2 .¤¹p|h. Îą > h > 5. p > 5Â…Îą > 5ž, pÎąâˆ’2 |h2 6 Îą2 ´Ă˜ÂŒU , Ă?Â?džkĂ˜ ÂŞpÎąâˆ’2 > Îą2 .

y3 Ă„S(n) = 3, džn = 3½Ü6.² yn = 3´Â?§Z2 (n) + 1 = S(n) ˜

‡); S(n) = S(32 ) = 6ž, n = 9, 18, 36, 45, ² y§Â‚Ă‘Ă˜ávÂ?§Z2(n) + 1 = S(n); S(n) = S(33 ) = 9ž, n = 33 , 2 ¡ 33 , 22 ¡ 33 , 5 ¡ 33 , 7 ¡ 33 , 23 ¡ 33 , 10 ¡ 33 , 14 ¡ 33 , 16 ¡ 33 , 20 ¡ 33 . dž² y n = 33 , 2 ¡ 33 , 22 ¡ 33 , 23 ¡ 33 , 24 ¡ 33 ávÂ?§Z2 (n) + 1 = S(n);Ă“ ÂŒ¹íĂ‘ S(n) = S(34 ) = 9ž, Â?k n = 34 , 2 ¡ 34 , 22 ¡ 34 , 23 ¡ 34 , 24 ¡ 34 ávÂ?§Z2 (n) + 1 = S(n). Â

Ă„S(n) = S(2Îą ).w,n = 1, 2Ă˜ávÂ?§Z2 (n) + 1 = S(n). eS(n) = S(4) =

4, @on = 4, 12.² yn = 4, 12ávÂ?§Z2(n) + 1 = S(n); S(n) = S(23 ) = 4ž, d žn = 8, 24.² yĂš nĂžĂ˜ávÂ?§Z2(n) + 1 = S(n); S(n) = S(24 ) = 6ž, d žn = 16, 3 ¡ 16, 5 ¡ 16, 15 ¡ 16.²u §Â‚ĂžĂ˜ávÂ?§Z2(n) + 1 = S(n); S(n) =

S(25 ) = 8ž,

n = 25 , 3 ¡ 25 , 5 ¡ 25 , 7 ¡ 25 , 15 ¡ 25 , 21 ¡ 25 , 35 ¡ 25 , 105 ¡ 25 , džN´ y§Â‚ĂžĂ˜ávÂ?§Z2 (n) + 1 = S(n); S(n) = S(2Îą ) = 2h, Îą > max{6, h} ž, n = m ¡ 2Îą , KS(m) < S(2Îą )Â…(m, 2) = 1.dženávÂ?§Z2 (n) + 1 = S(n), K dŸêZ2 (n) ½Ă‚,

n=m¡2

Îą (2h

− 1)2 (2h)2 = (2h − 1)2 h2 , 4

ddĂ­Ă‘2Îą |h2 6 (Îą − 1)2 < Îą2 , ĂšÂ‡Ă˜ ÂŞ9 Ă˜5´Ă˜ÂŒU , Ă?Â?A^êÆ8B{ N´y² Îą > 6ž,2Îą > (Îą − 1)2 > h2 .

nĂœ¹ÞÂˆÂŤÂœš, ĂĄÂ? ¤½n3.8 y².

50

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

3.8 'uSmarandache LCMŸê†SmarandacheŸê ފ ĂŠ?Âż ĂŞn,Ă?Âś SmarandacheŸêS(n)½Ă‚Â? êmÂŚ n|m!.= Ă’´S(n) = min{m : m ∈ N, n|m!}, Smarandache LCMŸêSL(n)½Ă‚Â? êk,ÂŚ n|[1, 2, ¡ ¡ ¡ , k],[1, 2, ¡ ¡ ¡ , k] LÂŤ1, 2, ¡ ¡ ¡ , k  ú ĂŞ. 'uÚ߇Ÿê

5Â&#x;, NþÆÜ?1 ĂŻĂ„, Âż Ă˜ ­Â‡ (J[8,14,29,37−39] . ~X, Šz[39]ĂŻ Ă„SL(n) ŠŠĂ™ÂŻK, y² ĂŹCúª X

n6x

2 (SL(n) − P (n))2 = ¡ Îś 5

5

5 x2 ¡ 2 ln x

!

5

+O

x2 ln x

!

,

Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f. Šz[39]„?Ă˜ Â?§SL(n) = S(n) ÂŒ)5, Âż )Ăť TÂŻK. =y² :? Ă›ávTÂ?§ ĂŞÂŒLÂŤÂ?n = 12½Ün = pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r p, Ù¼p1 , p2 , ¡ ¡ ¡ , pr , p´ Ă˜Ă“ ÂƒĂŞÂ…Îą1 , Îą2 , ¡ ¡ ¡ , Îąr , ´ávp > pÎąi i (i = 1, 2, ¡ ¡ ¡ , r) ĂŞ. !ĂŒÂ‡0 |^Ă? 9|ĂœÂ?{ĂŻĂ„¡ĂœĂžÂŠ X S(n) n6x

SL(n)

(3-18)

ĂŹC5Â&#x;. ĂšÂ˜ÂŻK´k¿Â , Ă?Â?ÂŞ(3-18) ĂŹC5‡N Ú߇ŸêŠŠĂ™ 5Æ5, XJĂŹCúª X S(n) âˆźx SL(n)

n6x

¤å, @oĂ’ÂŒ¹ä½ŸêS(n)ĂšSL(n) ŠA ??ƒ ! ! ĂŠĂšÂ˜ÂŻK?1 ĂŻ Ă„, Âży² § (5. äN/`=y² e¥ß‡(Ă˜. ½ n 3.9 ĂŠ?¿¢êx > 1kĂŹCúª X S(n) x ln ln x =x+O . SL(n) ln x n6x

½ n 3.10 Ê?¿¢êx > 1kÏCúª X P (n) x ln ln x =x+O , SL(n) ln x n6x

Ù¼P (n)LÂŤn  ÂŒÂƒĂ?f. w,½nÂĽ Ă˜ ‘´Âš~f , =Ă˜ Â‘Â†ĂŒÂ‘= ˜‡ lnlnlnxx Ă?f, ´Ă„ 3˜ ‡ r ĂŹCúªÂ?´Â˜Â‡k ÂŻK.

51

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

eÂĄy²Ú߇½n. Â?y²½n3.9, aq/, Â?ÂŒ¹íĂ‘½n3.10. ¯¢Þ²L{ ĂźC/ĂĄÂ? X S(n) X S(n) − SL(n) X = + 1 SL(n) SL(n) n6x n6x n6x   X |SL(n) − S(n)| . =x + O  SL(n)

(3-19)

n6x

y3|^ŸêS(n)9SL(n) 5Â&#x;Âą9Ă? †|ĂœÂ?{5 OÂŞ(3-19)ÂĽ Ă˜ ‘. dSL(n) 5Â&#x; n IOŠ)ÂŞÂ?pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k žk SL(n) = max{pÎą1 1 , pÎą2 2 , ¡ ¡ ¡ , pÎąk k }. eSL(n)Â?ÂƒĂŞp,@oS(n)Â?Â?ÂƒĂŞp.Ă?d, 3ڍÂœšekSL(n) − S(n) = 0.¤¹3 ÂŞ(3-19) Ă˜ ‘¼, ¤kš"‘7Ă‘y3@ ÂŚSL(n)Ă˜ uÂƒĂŞ ĂŞnÂĽ, = SL(n) = max{pÎą1 1 , pÎą2 2 , ¡ ¡ ¡ , pÎąk k } ≥ pÎą , Îą > 2. AÂ?ÂŤm[1, x]¼¤kávÞª^‡n 8Ăœ, ĂŠ?Âżn ∈ A, n = pÎą1 1 , pÎą2 2 , ¡ ¡ ¡ , pÎąk k =

pÎą ¡ n1 , Ù¼(p, n1 ) = 1. y3ŠßÂœš?Ă˜: A = B + C,Ù¼n ∈ BXJSL(n) =

pÎą >

ln2 x . 9(ln ln x)2

n ∈ CXJSL(n) = pÎą <

X |SL(n) − S(n)|

n6x

SL(n)

ln2 x . 9(ln ln x)2

u´k

X |SL(n) − S(n)| X |SL(n) − S(n)| + SL(n) SL(n) n∈B n∈C X X X 6 1+ 1 ≥ R1 + R2 .

=

n6

9x(ln ln x)2 ln2 x

ln2 x x 6pÎą 6 n 9(ln ln x)2 Îą>2

(3-20)

n∈C

y3ŠO OÂŞ(3-20)ÂĽ ˆ‘. Ă„k OR1. 5Âż pÎą 6 ln4 xžkÎą 6 4 ln ln x u´ dƒê½nk R1 6

X

x pÎą 6 n Îą>2

X

X

n6 lnx4 x

≪ ≪ ≪

n6 lnx4 x

X

n6 lnx4 x

X

r

X

x 9x(ln ln x)2 pÎą 6 n x 6n6 ln2 x Îą>2 ln4 x

X

√ x Îą6ln x

p6

X

1+

n

X

1+

x ln x + n ln ln x

ln x)2 x 6n6 9x(ln ln4 x ln2 x

X

ln x)2 x 6n6 9x(ln ln4 x ln2 x

x ln ln x . ln x

r

X

X

√ x Îą64 ln ln x

p6

n

1

x n (3-21)

52

1nĂ™ †SmarandacheŸê9–SmarandacheŸêƒ' ˜ Ă™§Ÿê

y3 OR2 ,5Âż 8ĂœCÂĽÂ?š ƒ Â‡ĂŞĂ˜ÂŹÂ‡L ĂŞpÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąk k ‡ê, Ă™ ÂĽÎąi 6 2 ln ln x, pi 6

ln x ,i 3 ln ln x

= 1, 2, ¡ ¡ ¡ . u´5Âż ƒêŠĂ™úª X y 1 1 ln p = y + O , − ln 1 − âˆź , ln y ln p ln p p6y

k R2 =

X

Y

16

n∈C

ln x p6 3 ln ln x

=

Y

ln x p6 3 ln ln x

 

X

06Îą62 ln ln x

p2 ln ln x 1 − ln1p



pι 

  −1 Y X 1 ≪ 1− exp 2 ln ln x ln p ln p ln x ln x p6 3 ln p6 3 ln ln x ln x  X 3 1  ≪ exp  ln x + 4 ln p ln x

x ≪ , ln x

p6 3 ln ln x

(3-22)

Ù¼exp(y) = ey . (ĂœÂŞ(3-20),(3-21)9ÂŞ(3-22), Ă­Ă‘ OÂŞ X |SL(n) − S(n)| x ln ln x ≪ . SL(n) ln x

(3-23)

n6x

|^ÂŞ (3-19)9ÂŞ(3-23)ĂĄÂ?Ă­Ă‘ĂŹCúª X S(n) x ln ln x =x+O . SL(n) ln x

n6x

u´ ¤ ½n3.9 y². 5Âż SL(n) = pÂ?ÂƒĂŞÂž,S(n) = P (n) = p; SL(n)Ă˜Â?ÂƒĂŞÂž, P (n) 6 S(n) 6 SL(n), u´dy²½n3.9 Â?{ĂĄÂ?Ă­Ă‘½n3.10.

3.9 'uSmarandache LCMŸê9Ă™ÊóŸê ĂŠu?Âż ĂŞn, Smarandache LCMŸê ½Ă‚Â? SL(n) = min{k : k|[1, 2, ¡ ¡ ¡ , k], k ∈ N}.

53

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

'u§ 5Â&#x;9ŸêÂ?§kNþÆÜ?1LĂŻĂ„, ˜ (J. ~XMurthy3 Šz[14]ÂĽy² nÂ?ÂƒĂŞÂž,SL(n) = S(n). Ӟ„?Ă˜

SL(n) = S(n), S(n) 6= n ÂŒ ) 5. Ăš pS(n) = min{m : n|m!, m ∈ N}Â?SmaranacheÂź ĂŞ. ƒ

Le Mao-

hua3Šz[19] )Ăť Murthy3Šz[14]JĂ‘ ÂŻK, Ă‘ Â…= n = 12 ½Ün =

pÎą1 1 pÎą2 2 ¡ ¡ ¡ pÎąr r p, (p > pÎąi i , i = 1, 2, ¡ ¡ ¡ , r)žÂ?§k).

Lv Zhongtian 3Šz[20]?Ă˜ SL(n) ĂŹC5Â&#x;, ˜‡ Ă? ފOÂŽĂş

ÂŞ X

n6x

k

X ci ¡ x2 Ď€ 2 x2 SL(n) = +O + 12 ln x i=2 lni x

x2 lnk+1 x

,

Ăšpx > 1Â?¢ê, kÂ? ĂŞ,cÂ?ÂŒOÂŽ ~ĂŞ. ĂŠu?Âż ĂŞn,Ă?Âś Smarandache LCMŸê ÊóŸê½Ă‚Â? SL∗ (n) = max{k : [1, 2, ¡ ¡ ¡ , k]|k, k ∈ N}. dSL∗ (n) ½Ă‚ÂŒÂąN´íĂ‘,SL∗ (1) = 1, SL∗ (2) = 2, SL∗ (3) = 1, SL∗ (4) = 2, SL∗ (5) = 1, SL∗ (6) = 3, SL∗ (7) = 1, SL∗ (8) = 2, SL∗ (9) = 1, SL∗ (10) = 2, . w,, nÂ?Ă› êž, SL∗ (n) = 1, nÂ?óêž, SL∗ (n) > 2. 'uڇŸê ٌ5Â&#x;, NþÆÜÂ?? 1LĂŻĂ„, ˜X ĂŻĂ„¤J, „Šz[4, 19 − 21].

~X3Šz[21]ÂĽ, Tian ChengliangĂŻĂ„ ŸêÂ?§ X SL∗ (d) = n d|n

Ăš X

SL∗ (d) = φ(n)

d|n

ÂŒ)5, Âż Ă‘cĂśÂ?kÂ?˜ ĂŞ)n = 1,

Ăś ĂŞ)Â?n = 1, 3, 14. d

3Šz[21]ÂĽ, Œ„ïÄ SL∗ (n) ?ĂŞĂš9ފ, ˜‡k (J: ∞ X SL∗ (n) n=1

ns

= Îś(s)

∞ X X (pÎąâˆ’1 )(ps − 1) Îą=1

p

[1, 2, ¡ ¡ ¡ , pι ]

Ăš X

n6x

SL∗ (n) = c ¡ x + o(ln2 x),

Ù¼Μ(s)Â?Riemann zeta-Ÿê. 54

1nÙ Smarandache¼ê9 Smarandache¼ê ' Ù§¼ê

Z3©z[59]ïÄ § (1)n Ûê;

P

d|n SL

∗

(d) =

P

d|n S

∗

(d), ¿ ÑÙ)

(2) 3†n , Kn = 2α pα1 1 pα2 2 · · · pαk k , p1 > 5, α > 1, αi > 0, k > 1, i = 1, 2, · · · , k;

(3) 3|n ,Kn = 2α · 3α1 pα2 2 · · · pαk k , p2 > 5, α > 1, αi > 0, k > 2, i = 1, 2, · · · , k.

   X  X ∗ ∗ SL (d) = S (d), n ∈ N , A= n:   d|n

½Â

d|n

∞ X 1 f (s) = , (Re(s) > 1), ns n=1 n∈A

K Z 'uÙ) 8Ü ð ª 1 f (s) = ζ(s) 1 − s , 12 Ù¥ζ(s) Riemann zeta-¼ê. ! Ì 8 ´|^Ð êØÚ©Û {ïÄ¼ê § Y SL∗ (d) + 1 = 2ω(n)

(3-24)

d|n

)5, ¿ Ù¤k ê), =Ò´e¡½n. ½ n 3.11 é?¿ ên, ¼ê §(3-24)k) = n = pα , α > 1, p > 3 ê. y ² : dSL∗ (n) ½Â, ´ n = 1, 2w,Ø´ §(3-24) ). e¡©ü« 6?1? Ø: (1)en = p > 3 ê, Kω(n) = ω(p) = 1,

Q

SL∗ (d)+1 = SL∗ (1)·SL∗ (p)+1 = 2,

d|n

¤±, w,n = p ê´ §(3-24) ). (2)en Üê, @o Xe©Û:

1) n = 2α , α > 1 ,´ ω(n) = ω(2α ) = 1, Y SL∗ (d) + 1 = SL∗ (1) · SL∗ (2) · · · SL∗ (2α ) + 1 = 2α + 1, d|n

K2α + 1 = 2,=2α = 1, α = 0,ù α > 1gñ. ¤±n = 2α ,α > 1Ø´(3-24) ). 2) n = 2α pβ ,α > 1, β > 1, p > 3 ê, d ω(n) = ω(2α pβ ) = 2, 55

Smarandache ¼ê9Ù '¯KïÄ

dSL∗ (n) ½ÂÚ5 ´

Q

SL∗ (2m pn ) + 1 > 2α+β + 1 > 4,

m=0 n=0

d|2α pβ α β

β α Q Q

SL∗ (d) + 1 =

n = 2 p , α > 1, β > 1, p > 3 , §(3-24)Ã).

3) n = 2α pα1 1 pα2 2 · · · pαk k , α > 1, αi > 1, i = 1, 2, · · · , k, k > 1, pi > 3 pÉ

ê.´ ω(n) = ω(2α pα1 1 pα2 2 · · · pαk k ) = k + 1, Y

α

∗

SL (d) + 1 = α

α

α Y α1 Y

n=0 m1 =0

d|2α p1 1 p2 2 ···pk k

>2

α+

k P

αi

i=1

···

αk Y

mk =0

mk 1 m2 SL∗ (2n pm 1 p2 · · · pk ) + 1

+1

> 2α+k·1 + 1 > 2k+1 + 1 > 2k+1 . ¤±n = 2α pα1 1 pα2 2 · · · pαk k , §(3-24) Ã).

4) n = pα , α > 1, p > 3 ê. á kω(n) = ω(pα ) = 1, d ½  α Q Q SL∗ (pm ) + 1 = 1 + 1 = 2, ¤±(3-24)ª¤á. n = pα ´ SL∗ (d) + 1 = m=0

d|pα

§(3-24) ).

5) n = pα1 1 pα2 2 · · · pαk k , αi > 1, 3 6 p1 < p2 · · · < pk , i = 1, 2, · · · , k, k > 2, du ω(n) = ω(pα1 1 pα2 2 · · · pαk k ) = k, α

Y α

∗

SL (d) + 1 = α

d|p1 1 p2 2 ···pk k

α1 α2 Y Y

m1 =0 m2 =0

···

αk Y

mk =0

mk 1 m2 SL∗ (pm 1 p2 · · · pk ) + 1 = 1 + 1 = 2,

d 7k2 = 2k ,=k = 1,ùÚk > 2gñ.¤±n = pα1 1 pα2 2 · · · pαk k , §(3-24) Ã).

nܱþ?Ø(1)Ú(2) , §(3-24)k) = n = pα , α > 1, p > 3 ê,

ù Ò ¤ ½n3.11 y².

56

1oĂ™ SmarandacheS ĂŻĂ„

1oĂ™ SmarandacheS ĂŻĂ„ 4.1 Smarandache LCM 'ÇS I (x1 , x2 , ..., xt )Ăš[x1 , x2 , ..., xt ]ŠOLÂŤx1 , x2 , ..., xt  ÂŒĂşĂ?fĂš ú ĂŞ. r ´Â˜Â‡ÂŒu1 ĂŞ. ĂŠ?Âż ĂŞn, K T (r, n) =

[n, n + 1, ..., n + r − 1] [1, 2, ..., r]

PŠSLR(r) = {T (r, n)}∞ n=1 ÂĄÂ? Â?r Smarandache LCM'ÇS . Maohua Le

ĂŻĂ„ § 5Â&#x;, ‰Ñ 'uSLR(2), SLR(3) ĂšSLR(4) 4íúª, ʲw, 1 T (2, n) = n(n + 1). 2   1 n(n + 1)(n + 2), en ´Ă›ĂŞ ; T (3, n) = 6  1 n(n + 1)(n + 2), en ´óê. 12   1 n(n + 1)(n + 2)(n + 3), en 6≥ 0(mod3); T (4, n) = 24  1 n(n + 1)(n + 2)(n + 3), en ≥ 0(mod3). 72

!‰Ñ r = 5ž 4íúª, =‰ÑeÂĄ ½n. ½ n 4.1 ĂŠu?Âż ĂŞn,k T (5, n) =

57

[22]

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

 1  n(n + 1)(n + 2)(n + 3)(n + 4),   120         1    240 n(n + 1)(n + 2)(n + 3)(n + 4),          1   360 n(n + 1)(n + 2)(n + 3)(n + 4),    

n 6≥ 0(mod2)…n 6≥ 0(mod3)…n 6≥ 0(mod4)

…n + 1 6≥ 0(mod3);

n ≥ 0(mod2)…n 6≥ 0(mod3)…n 6≥ 0(mod4)

…n + 1 6≥ 0(mod3);

n ≥ 0(mod2), n 6≥ 0(mod3)½n 6≥ 0(mod2),

n 6≥ 0(mod3), n 6≥ 0(mod4), n + 1 6≥ 0(mod3);

1  n(n + 1)(n + 2)(n + 3)(n + 4), n 6≥ 0(mod3), n ≥ 0(mod4), n + 1 6≥ 0(mod3);   480    1   720 n(n + 1)(n + 2)(n + 3)(n + 4), n ≥ 0(mod2), n 6≥ 0(mod3), n 6≥ 0(mod4),      n + 1 ≥ (mod3)½n ≥ 0(mod2), n ≥ 0(mod3),      n 6≥ 0(mod4);      1   1440 n(n + 1)(n + 2)(n + 3)(n + 4), n 6≥ 0(mod3), n ≥ 0(mod4), n + 1 ≥ 0(mod3)     ½n ≥ 0(mod3), n ≥ 0(mod4).

eÂĄ|^ ÂŒĂş ĂŞĂš ú ĂŞ 5Â&#x;5y²½n4.1. Â?,[n, n + 1, n + 2, n +

3, n + 4] = [n(n + 1), (n + 2)(n + 3), n + 4], ´[n(n + 1), (n + 2)(n + 3), n + 4](n(n + 1), (n+2)(n+3), n+4) 6= n(n+1)(n+2)(n+3)(n+4). ¤¹Êu?¿ß‡ ĂŞ  ÂŒĂş

ꆠú ĂŞ 5Â&#x;Ă­2 ?Âżn‡9n‡¹Þ ĂŞÂžÂżĂ˜¤å. ÂŚ)T (r, n) 4íúª, 'Â…´Â‡ÂŚĂ‘[n, n + 1, n + 2, n + 3, n + 4], 3dònŠOŠa?1`². du Ă“{'X´ d'X, ¤¹^§ÂŒÂąr ĂŞ8yŠÂ?eZ‡ da. (1)en ≥ 0(mod2), n 6≥ 0(mod3), n 6≥ 0(mod4), [n, n +1, n +2, n +3, n +4] Ăł

Ăł

Ăł

2

3

4

5

6

10

11

12

13

14

14

15

16

17

18

22

23

24

25

26

26

27

28

29

30

34

35

36

37

38

38

39

40

41

42

ƒ nóê[n, n+2, n+4] = 14 [n(n+2)(n+4)], ƒ Ûê[n+1, n+3] = (n+1)(n+3), en + 1 ≥ 0(mod3)ž,Kn + 4 ≥ 0(mod3),¤¹   1 [n(n + 1)(n + 2)(n + 3)(n + 4)], n + 1 ≥ 0(mod3)ž, [n, n + 1, n + 2, n + 3, n + 4] = 12  1 [n(n + 1)(n + 2)(n + 3)(n + 4)], n + 1 6≥ 0(mod3)ž. 4

58

1oĂ™ SmarandacheS ĂŻĂ„

(2)en ≥ 0(mod3),n 6≥ 0(mod2), [n, n +1, n +2, n +3, n +4] ó

Ăł

3

4

5

6

7

9

10

11

12

13

15

16

17

18

19

21

22

23

24

25

27

28

29

30

31

33

34

35

36

37

Ăź ƒ Ăł ĂŞ[n + 1, n + 3] =

1 [(n 2

+ 1)(n + 3)], q � �n ≥ 0(mod3),n + 3 ≥

0(mod3),¤¹[n, n + 3] = 13 [n(n + 3)],¤¹dž

1 [n, n + 1, n + 2, n + 3, n + 4] = [n(n + 1)(n + 2)(n + 3)(n + 4)]. 6 (3)en ≥ 0(mod4),n 6≥ 0(mod3), [n, n +1, n +2, n +3, n +4] ó

Ăł

Ăł

4

5

6

7

8

8

9

10

11

12

16

17

18

19

20

20

21

22

23

24

28

29

30

31

32

32

33

34

35

36

40

41

42

43

44

nƒ óê, qĂ?Â?n ≥ 0(mod4),n + 4 ≥ 0(mod4), ¤¹[n, n + 2, n + 4] = 18 [n(n +

2)(n + 4)], en + 1 ≥ 0(mod3),Kn + 4 ≥ 0(mod3),¤¹   1 n(n + 1)(n + 2)(n + 3)(n + 4), n + 1 6≥ 0(mod3)ž, [n, n + 1, n + 2, n + 3, n + 4] = 8  1 n(n + 1)(n + 2)(n + 3)(n + 4), n + 1 ≥ 0(mod3)ž. 24

(4)en ≥ 0(mod2),n ≥ 0(mod3),n 6≥ 0(mod4),

59

Smarandache ¼ê9Ù '¯KïÄ

[n, n +1, n +2, n +3, n +4] ó

ó

ó

6

7

8

9

10

18

19

20

21

22

30

31

32

33

34

42

43

44

45

46

54

55

56

57

58

n óê[n, n + 2, n + 4] = 14 [n(n + 2)(n + 4)], n ≡ n + 3 ≡ 0(mod3),¤± [n, n + 1, n + 2, n + 3, n + 4] =

1 [n(n + 1)(n + 2)(n + 3)(n + 4)]. 12

(5)en ≡ 0(mod3),n ≡ 0(mod4), [n, n +1, n +2, n +3, n +4] ó

ó

ó

12

13

14

15

16

24

25

26

27

28

36

37

38

39

40

48

49

50

51

52

60

61

62

63

64

n óê[n, n + 2, n + 4] = 18 [n(n + 2)(n + 4)],Ï , n ≡ n + 4 ≡ 0(mod4),n ≡

n + 3 ≡ 0(mod3), ¤±[n, n + 3] = 13 [n(n + 3)],¤± [n, n + 1, n + 2, n + 3, n + 4] =

1 [n(n + 1)(n + 2)(n + 3)(n + 4)]. 24

(6)en 6≡ 0(mod2),n 6≡ 0(mod3),n 6≡ 0(mod4)

60

1oĂ™ SmarandacheS ĂŻĂ„

[n, n +1, n +2, n +3, n +4] Ăł

Ăł

5

6

7

8

9

7

8

9

10

11

11

12

13

14

15

13

14

15

16

17

17

18

19

20

21

19

20

21

22

23

23

24

25

26

27

25

26

27

28

29

ĂźÂƒ óê[n + 1, n + 3] = 12 [(n + 1)(n + 3)], en + 1 ≥ 0( mod 3),[n, n + 1, n + 2, n +

3, n+4] = 16 [n(n+1)(n+2)(n+3)(n+4)]; en+1 ≥ 0( mod 3), [n, n+1, n+2, n+3, n+4] = 1 2 [n(n

+ 1)(n + 2)(n + 3)(n + 4)].

r¹Þ8a?1 n, ÂƒĂ“^‡ 4íúª8Â?˜a, B ½n4.1.

4.2 Smarandache LCM 'ÇS II ĂžÂ˜!‰Ñ Smarandache LCM'ÇS ŠO3 u2,3,4,5ž 4íúª, ĂšÂ˜ !òïĂ„Smarandache LCM'ÇS ˜„Ă?‘úª, ‰ÑSmarandache LCM'ÇS ŠO'u r!'un ˜„Ă?‘úª, = eÂĄ A‡(J. ½ n 4.2 ĂŠ?Âżg,ĂŞn, r§¡Â‚kXe4íúª¾ T (r + 1, n) =

([1, 2, ..., r], r + 1) n+r T (r, n), r + 1 ([n, n + 1, ..., n + r − 1], n + r)

Ă­ Ă˜ 4.1 XJr + 1 Ăšn + r Ă‘´Âƒê§K¡Â‚kÂ˜Â‡ÂƒĂŠ{Ăź úª¾ T (r + 1, n) =

n+r T (r, n). r+1

½ n 4.3 ĂŠ?Âżg,ĂŞn, r§¡Â‚k,˜‡4íúª¾ T (r, n + 1) =

n+r (n, [n + 1, ..., n + r]) T (r, n). n ([n, n + 1, ..., n + r − 1], n + r)

Ă­ Ă˜ 4.2 XJn Ăšn + r Ă‘´ÂƒĂŞÂ…r < n§K¡Â‚Â?k˜‡Â?{Ăź úª¾ T (r, n + 1) =

n+r ¡ T (r, n); n

61

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

XJn Ăšn + r Ă‘´ÂƒĂŞÂ…r > n§K¡Â‚k T (r, n + 1) = (n + r) ¡ T (r, n). ½ n 4.4 ĂŠ?Âżg,ĂŞn, r§¡Â‚k4íúª¾ T (r + 1, n + 1) =

([1, 2, ..., r], r + 1) n+r n+r+1 ¡ ¡ n r+1 ([n + 1, ..., n + r], n + r + 1) (n, [n + 1, ..., n + r]) ¡ ¡ T (r, n). ([n, n + 1, ..., n + r − 1], n + r)

Â? y²ÚA‡½n, I‡¹eA‡Ún. Ăš n 4.1 ĂŠ?Âż ĂŞaĂšb, k(a, b)[a, b] = ab. Ăš n 4.2 ĂŠ?Âż ĂŞs, tÂ…s < t, k (x1 , x2 , ..., xt ) = ((x1 , ..., xs ), (xs+1 , ..., xt )) Ăš [x1 , x2 , ..., xt ] = [[x1 , ..., xs ], [xs+1 , ..., xt ]]. Ăšn4.1Ăš4.2 y²ÂžĂŤ Šz[6]. eÂĄ5y²½n4,2. Ă„kŠâŠâT (r, n) ½Ă‚, Ăšn4.1Ăš4.2, k [n, n + 1, ..., n + r] = [[n, n + 1, ..., n + r − 1], n + r] [n, n + 1, ..., n + r − 1](n + r) = , ([n, n + 1, ..., n + r − 1], n + r) [1, 2, ¡ ¡ ¡ , r + 1] = [[1, 2, ¡ ¡ ¡ , r], r + 1] =

(r + 1)[1, 2, ¡ ¡ ¡ , r] , ([1, 2, ¡ ¡ ¡ , r], r + 1)

u´§¡Â‚ 'uT (r + 1, n) 4íúª [n, n + 1, ..., n + r] [1, 2, ..., r + 1] [[n, n + 1, ..., n + r − 1], n + r] = [[1, 2, ..., r], r + 1]

T (r + 1, n) =

=

(n+r)[n,n+1,...,n+r−1] ([n,n+1,...,n+r−1],n+r) (r+1)[1,...,r] ([1,2,...,r],r+1)

n + r [n, n + 1, ..., n + r − 1] ([1, 2, ..., r], r + 1) r+1 [1, 2, ..., r] ([n, n + 1, ..., n + r − 1], n + r) n+r ([1, 2, ..., r], r + 1) = T (r, n). r + 1 ([n, n + 1, ..., n + r − 1], n + r) =

Ăš Ă’ ¤ ½n4.2 y². 62

1oĂ™ SmarandacheS ĂŻĂ„

Ă­Ă˜4.1 y². r + 1 Ăšn + r Ă‘´ÂƒĂŞÂž, w,k ([1, 2, ¡ ¡ ¡ , r], r + 1) = 1, ([n, n + 1, ¡ ¡ ¡ , n + r − 1], n + r) = 1, Kd½n4.2, ¡Â‚k ([1, 2, ..., r], r + 1) n+r T (r, n) r + 1 ([n, n + 1, ..., n + r − 1], n + r) n+r = T (r, n). r+1

T (r + 1, n) =

ÚÒy² Ă­Ă˜4.1. ½n4.3 y²: Ă“ ŠâT (r, n) ½Ă‚ÚÚn4.1†Ún4.25y²½n4.3. Ă„kd Ăšn4.1ÚÚn4.2, k [n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]] =

n[n + 1, n + 2, ¡ ¡ ¡ , n + r] , (n, [n + 1, n + 2, ¡ ¡ ¡ , n + r])

u´ [n + 1, n + 2, ¡ ¡ ¡ , n + r] [n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]] ¡ (n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]) = n [n, n + 1, n + 2, ¡ ¡ ¡ , n + r] ¡ (n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]) = , n u´ Ă‘'uT (r, n + 1) 4íúª [n + 1, ..., n + r] [1, 2, ..., r] [n, n + 1, ..., n + r](n, [n + 1, ..., n + r]) 1 = n [1, 2, ..., r] (n, [n + 1, ..., n + r]) [n, n + 1, ..., n + r − 1](n + r) = n[1, 2, ..., r] ([n, n + 1, ..., n + r − 1], n + r) n+r (n, [n + 1, ..., n + r]) [n, n + 1, ..., n + r − 1] = n ([n, n + 1, ..., n + r − 1], n + r) [1, 2, ..., r] n+r (n, [n + 1, ..., n + r]) = T (r, n). n ([n, n + 1, ..., n + r − 1], n + r)

T (r, n + 1) =

ÚÒ ¤ ½n4.3 y². Kk

Ă­Ă˜4.2 y². n ´ÂƒĂŞÂ…r < n ž, [n + 1, n + 2, ¡ ¡ ¡ , n + r] 7Ă˜škƒĂ?fn§ (n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]) = 1;

63

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ӟ, en + r Â?´ÂƒĂŞ, K7,k ([n, n + 1, ¡ ¡ ¡ , n + r − 1], n + r) = 1. Ă?dÂŒÂą T (r, n + 1) =

n+r (n, [n + 1, ..., n + r]) n+r T (r, n) = T (r, n). n ([n, n + 1, ..., n + r − 1], n + r) n

n´ÂƒĂŞÂ…r > nž, [n + 1, n + 2, ¡ ¡ ¡ , n + r] 7šÂƒĂ?fn, Ă?d (n, [n + 1, n + 2, ¡ ¡ ¡ , n + r]) = n, Ӟdun + r ´ÂƒĂŞ, Ă?dk (n, [n + 1, ..., n + r]) n+r T (r, n) n ([n, n + 1, ..., n + r − 1], n + r) = (n + r)T (r, n).

T (r, n + 1) =

ÚÒy² Ă­Ă˜4.2. ½n4.4 y²: A^½n4.2Ăš½n4.3 ĂŠN´ ½n4.4 (J, = T (r + 1, n + 1) ([1, 2, ..., r], r + 1) n+r+1 T (r, n + 1) = r + 1 ([n + 1, ..., n + r], n + r + 1) (n + r + 1)(n + r) ([1, 2, ..., r], r + 1) (n, [n + 1, ..., n + r]) = T (r, n). (r + 1)n ([n + 1, ..., n + r], n + r + 1) ([n, ..., n + r − 1], n + r) ÚÒ ¤ ½n4.4 y².

4.3 Smarandache 1 ÂŞ 4.3.1 SmarandacheĂŒÂ‚1 ÂŞ ĂŠu?Ă› ĂŞn,n Ă— n1 ÂŞ

1

2

.

..

n − 1

n

3 ¡¡¡ n 1

.. .. ..

. . .

n ¡ ¡ ¡ n − 3 n − 2

1 ¡¡¡ n − 2 n − 1

2 ¡¡¡ n − 1

ÂĄÂ?n SmarrandacheĂŒÂ‚1 ÂŞ, PÂ?SCND(n).

64

n

(4-1)

1oĂ™ SmarandacheS ĂŻĂ„

a,d´EĂŞ,n Ă— n1 ÂŞ

a a+d ¡¡¡

a+d a + 2d ¡¡¡

.. ..

. .

a + (n − 2)d a + (n − 1)d ¡ ¡ ¡

a + (n − 1)d a ¡¡¡

a + (n − 2)d a + (n − 1)d

a + (n − 1)d a

.. ..

. .

a + (n − 4)d a + (n − 3)d

a + (n − 3)d a + (n − 2)d

(4-2)

ÂĄÂ?'uĂŞĂŠ(a, d) n SmarrandacheĂŒÂ‚ÂŽâ?ĂŞ1 ÂŞÂ…LÂŤÂ?SCAD(n; a, d). 3Šz[34]ÂĽ, Murthy‰Ñ e¥ß‡Ún. Ăš n 4.3 ĂŠu?Ă› ĂŞn, n

SCND(n) = (−1) 2 nn−1

(n + 1) . 2

(4-3)

Ăš n 4.4 ĂŠu?Âż ĂŞn†?ÂżEĂŞĂŠa, d, k   a, n = 1ž, SCAD(n; a, d) = n (a+(n−1)d)  (−1) 2 (nd)n−1 , n > 1ž.

(4-4)

Maohua|^Ž Š 1 ª

a1 a2 ¡ ¡ ¡ an−1

a a ¡ ¡ ¡ a n−2

n 1

. . ..

.. .. .

a3 a4 ¡ ¡ ¡ a1

a2 a3 ¡ ¡ ¡ an

(4-5)

2

3Šz[35]ÂĽ, Le Maohuay² 'uSCND(n)†SCAD(n; a, d) Ă&#x;Ăż. =Le

an an−1 .. . a2 a1

Y

= (a1 + a2 x + ¡ ¡ ¡ + an xn )

xn =1

y² Ăšn4.3†4.4¢SĂžÂŒÂąÂ† OÂŽ=ÂŒ.

X‡OÂŽÂŞ(4-1), Â?‡l . 1n1mŠ, ĂžÂ˜1 −1 \ e˜1, 1˜1 Ă˜Ă„, ÂŚ

1 2 ¡ ¡ ¡ n − 1 n

1 1 ¡ ¡ ¡ 1 1 − n

. . .. .. SCND(n) =

.. .. . .

1 1 ¡ ¡ ¡ 1 1

1 1 − n ¡ ¡ ¡ 1 1

65

.

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

,

12 1n Ăœ\ 1˜ , 2r1˜ Ă?m (Ă˜,=

n(n − 1)/2 2 ¡ ¡ ¡ n − 1 n

0 1 ¡ ¡ ¡ 1 1 − n

. . . . .. .. .. ..

SCND(n) =

0 1 ¡ ¡ ¡ 1 1

0 1 − n ¡¡¡ 1 1

1 ¡¡¡ 11 − n

.

. . .. ..

n(n + 1)

..

=

1 ¡¡¡ 1 1

2

1 − n ¡ ¡ ¡ 1 1

n−1

(n + 1) = (−1) nn−1 . 2 n 2

Ă“ , ÂŞ(4-2)Â?ÂŒ^ڍÂ?{OÂŽ. rSCND(n)˜„z, a1 , a2 , ¡ ¡ ¡ , an ´n‡EĂŞ,n Ă— n 1 ÂŞ

a a ¡ ¡ ¡ a a

1 2 n−1 n

a

a ¡ ¡ ¡ a a 2 3 n 1

.

. . .

..

.. .. ..

an−1 an ¡ ¡ ¡ an−3 an−2

an a1 ¡ ¡ ¡ an−2 an−1

(4-6)

ÂĄÂ?'uĂŤĂŞa1 , a2 , ¡ ¡ ¡ , an n SmarandacheĂŒÂ‚1 ÂŞÂ…LÂŤÂ?SCD(a1 , a2 , ¡ ¡ ¡ , an ).ĂŠ

un SmarandacheĂŒÂ‚1 ÂŞSCD(a1 , a2 , ¡ ¡ ¡ , an ), A^ÂŞ(4-5), ÂŒ eÂĄ ½n. ½ n 4.5 ĂŠun‡?ÂżEĂŞa1 , a2 , ¡ ¡ ¡ , an , SCD(a1 , a2 , ¡ ¡ ¡ , an ) = (−1)r

Y

(a1 + a2 x + ¡ ¡ ¡ + an xn−1 ),

xn =1

Ù¼   n − 1, en´óê, r= 2  (n − 1)/2, en´Ă›ĂŞ.

66

(4-7)

1oĂ™ SmarandacheS ĂŻĂ„

SCD(a1 , a2 , ¡ ¡ ¡ , an ) ,Â˜ÂŤAĂ?Âœš´: a, q߇EĂŞ, n Ă— n1 ÂŞ

n−2 n−1

a aq ¡ ¡ ¡ aq aq

aq

2 n−1 aq ¡ ¡ ¡ aq a

. . . .

..

.. .. ..

aq n−2 aq n−1 ¡ ¡ ¡ aq n−4 aq n−3

n−1 n−3 n−2

aq a ¡ ¡ ¡ aq aq

(4-8)

ÂĄÂ?'u(a, q) n SmarandacheĂŒÂ‚AĂ›?ĂŞ1 ÂŞÂ…LÂŤÂ?SCGD(n; a, q). ĂŠu, n SmarandacheĂŒÂ‚AĂ›?ĂŞ1 ÂŞSCGD(n; a, q), kXe ½n: ½ n 4.6 ĂŠu?Âż ĂŞn†?ÂżEĂŞĂŠa, q, SCGD(n; a, q) = (−1)r an (1 − q n )n−1 .

(4-9)

½n4.6 y²:lÂŞ(4-6)!(4-8)k SCGD(n; a, q) = SCD(a, aq, aq 2 , ¡ ¡ ¡ , aq n−1 ), 2dÂŞ(4-7), ÂŒÂą SCD(a, aq, aq 2 , ¡ ¡ ¡ , aq n−1 ) = (−1)r

Y

(a + aqx + aq 2 x2 + ¡ ¡ ¡ + aq n−1 xn−1 )

xn =1

r n

= (−1) a

Y

(1 + qx + q 2 x2 + ¡ ¡ ¡ + q n−1 xn−1 ).

xn =1

XJxn = 1,K(1 + qx + q 2 x2 + ¡ ¡ ¡ + q n−1 xn−1 )(1 − qx) = 1 − q n ,du Y Y 1 1 n n (1 − qx) = q − x = 1 − qn , −x =q n q q xn =1 xn =1 ÂŒ ÂŞ(4-9). ¤¹½n4.6¤å. ¢SĂž, Ăšn4.3!Ăšn4.4!½n4.6Ă‘´½n4.5 A Ă?Âœš.

4.3.2 Smarandache VĂŠÂĄ1 ÂŞ 3Šz[35]ÂĽ, ĂŠu?Âż ĂŞn, n Ă— n1 ÂŞ

2 ¡¡¡ n − 1 n

1

2 3 ¡¡¡ n n − 1

. . .. ..

.. .. . .

n − 1 n ¡ ¡ ¡ 3 2

n n − 1 ¡¡¡ 2 1

ÂĄÂ?n SmarandacheVĂŠÂĄ1 ÂŞÂ…LÂŤÂ?SBND(n). 67

(4-10)

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

a,bĂźEĂŞ,n Ă— n1 ÂŞ

a a + d ¡ ¡ ¡ a + (n − 2)d a + (n − 1)d

a+b

a + 2d ¡ ¡ ¡ a + (n − 1)d a + (n − 2)d

. . . .

.. .. .. ..

a + (n − 2)d a + (n − 1)d ¡ ¡ ¡ a + 2d

a + d

a + (n − 1)d a + (n − 2)d ¡ ¡ ¡ a + d

a

(4-11)

ÂĄÂ?'u(a, d) n SmarandacheVĂŠÂĄÂŽâ?ĂŞ1 ÂŞÂ…LÂŤÂ?SBAD(n; a, d). 'un SmarandachVĂŠÂĄ1 ÂŞSBND(n)†SmarandachVĂŠÂĄÂŽâ?ĂŞ1 ÂŞSBAD(n; a, d) , Le Maohuay² e Ăšn4.6,Ăšn4.7. Ăš n 4.5 ĂŠu?Âż ĂŞn, SBND(n) = (−1)

n(n−1) 2

2n−1 (n + 1).

(4-12)

Ăš n 4.6 ĂŠu?Âż ĂŞn†?ÂżEĂŞĂŠa, d, SBAD(n; a, d) = (−1)

n(n−1) 2

2n−2 dn−1 (2a + (n − 1)d) .

a, d´EĂŞ, n Ă— n1 ÂŞ

aq ¡¡¡

a

aq aq 2 ¡ ¡ ¡

. ..

.. .

aq n−2 aq n−1 ¡ ¡ ¡

n−1 n−2

aq aq ¡¡¡

aq aq

aq n−1 aq n−2

.. ..

. .

2

aq aq

aq a n−2

(4-13)

n−1

(4-14)

ÂĄÂ?'u(a, q) n SmarandacheVĂŠÂĄAĂ›?ĂŞ1 ÂŞÂ…LÂŤÂ?SBGD(n; a, q). 'un SmarandacheVĂŠÂĄAĂ›?ĂŞ1 ÂŞSBGD(n; a, q),keÂĄ ½n. ½ n 4.7 ĂŠu?Âż ĂŞn†EĂŞĂŠa, d,   0, n = 2ž, SCGD(n; a, d) = n(n−1)  (−1) 2 an q n(n−2)(n−1) (q 2 − 1)n−1 , n 6= 2ž. ½n4.7 y²:‡OÂŽÂŞ(4-14), Â?IOÂŽ

q ¡ ¡ ¡ an−2 q n−1

1

q 2 n−1 n−2

q ¡ ¡ ¡ q q

.

.. .. ..

..

. . . .

n−2 n−1 2

q

q ¡¡¡ q q

n−1 n−2

q q ¡¡¡ q 1

68

(4-15)

(4-16)

1oĂ™ SmarandacheS ĂŻĂ„

dÂŞ(4-14), n = 1½n = 2ž, ÂŒ ÂŞ(4-15)¤å. Ă?dÂŒb½n > 2. rÂŞ(4-16) 12 JĂ‘ĂşĂ?fq,12 −1 \ 11 , , U11 Ă?m,

n−2 n−1

1 q ¡ ¡ ¡ a q q ¡¡¡

1

q

q q 2 ¡ ¡ ¡ q n−1 q n−2

q3 ¡ ¡ ¡

.

.. .. .. ..

..

= q(−1)n+1 (q n−1 − q n−3 ) ¡ ... . . . .

q n−2 q n−1 ¡ ¡ ¡ q 2 q

q n−3 q n−1 ¡ ¡ ¡

n−1 n−2

n−2 n−2

q

q q ¡¡¡ q 1 q ¡¡¡

an−2 q n−1

q n−1 q n−2

.. ..

. .

q 3 q 2

q2 q

= q(−1)n−1 (q n−1 − q n−3 )q 2 (−1)n ¡ (q n−2 − q n−4 ) ¡ ¡ ¡

n−1

1q

q n−2 (−1)4 (q 2 − 1)

n−2

q q = (−1)

n(n−1) 2

q (n−2)(n−1) (q 2 − 1)n−1 .

dd, ÂŒ ÂŞ(4-15). ½n4.7 y. Ă­2SBND(n) ˜„œ/. a1 , a2 , ¡ ¡ ¡ , an ,´n ‡EĂŞ, n Ă— n1 ÂŞ

a2 ¡ ¡ ¡ an−1 an

a1

a a3 ¡ ¡ ¡ an an−1

2

.

.. .. ..

.. . . .

an−1 an ¡ ¡ ¡ a3 a2

an an−1 ¡ ¡ ¡ a2 a1

(4-17)

ÂĄÂ?'uĂŤĂŞa1 , a2 , ¡ ¡ ¡ , an n SmarandacheVĂŠÂĄ1 ÂŞÂ…LÂŤÂ?SBD(a1 , a2 , ¡ ¡ ¡ , an ). ĂŠuSBD(a1 , a2 , ¡ ¡ ¡ , an ) OÂŽ, „´Â˜Â‡Â™)Ăť ÂŻK.

4.4 Smarandache ĂŞ N + ´ N ĂŞ 8Ăœ. ĂŠu ĂŞn, S(n) = min{k|k ∈ N + , n|k!}.

(4-18)

Xd S(n)ÂĄÂ?'un SmarandacheŸê. n´ ĂŞ, XJn Ă˜Ă“ ĂŞÂƒĂš u2n, KÂĄn´ ĂŞ. Ă?Âą5, ĂŞÂ˜Â†´êĂ˜ÂĽ Â˜Â‡Ăš<'5 ÂŻK[41] .  C, Ashbacher[42] ò ĂŞ VgĂ­2 SmarandacheŸê‰Œ, òáv X S(d) = n + 1 + S(n)

(4-19)

d|n

ĂŞnÂĄÂ?Smarandache ĂŞ.ĂŠd, Ashbachery² : n 6 106 ž, =kSmarandache

69

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

ĂŞ12. !ò?Â˜Ăš0 Smarandache ĂŞ, =y² : ½ n 4.8 =kSmarandache ĂŞ12. Ăš n 4.7 XJ n = pr11 pr22 ¡ ¡ ¡ prkk

(4-20)

S(n) = max {S(pr11 ), S(pr22 ), ¡ ¡ ¡ , S(prkk )} .

(4-21)

´ ên IOŠ)ª, K

y ² :̈́Šz[43]. Ăš n 4.8 ĂŠuÂƒĂŞpĂš ĂŞr, 7kS(pr ) 6 pr. y ² :̈́Šz[43]. Ăš n 4.9 ĂŠu ĂŞn, d(n)´n Ă˜êŸê. dž,d(n)´Ăˆ5Ÿê; (4-20)´n I OŠ)ÂŞ,K d(n) = (r1 + 1)(r2 + 1) ¡ ¡ ¡ (rk + 1).

(4-22)

y ² :̈́Šz[44]ÂĽ ~6.4.2. Ăš n 4.10 Ă˜ ÂŞ n < 2, n ∈ N d(n)

(4-23)

=k)n = 1, 2, 3, 4, 6. eÂĄy²½n4.8. n´¡Ăœn 6= 12 Smarandache ĂŞ. ŠâŠz[41]ÂĽ (

J, ÂŒ n > 106 . (4-20)´n IOŠ)ÂŞ. ŠâĂšn4.7ÂŒ S(n) = S(pr ),

(4-24)

p = pj , r = rj , 1 6 j 6 k.

(4-25)

n|S(pr )!,

(4-26)

S(d) 6 S(pr ).

(4-27)

Ù¼

l(4-24)Œ

¤¹Êun ?Ă› ĂŞdĂ‘k

70

1oĂ™ SmarandacheS ĂŻĂ„

ĂŠu ĂŞn, g(n) =

X

S(d).

(4-28)

d|n

dž, l(4-19)Ăš(4-28)ÂŒ n¡Ăœ g(n) = n + 1 + S(n).

(4-29)

q d(n)´n Ă˜êŸê.l(4-27)Ăš(4-29)ÂŒ d(n)S(pr ) > n.

(4-30)

dul(4-20)Ăš(4-24)ÂŒ gcd(pr , n/pr ) = 1, ¤¹ n = pr m, m ∈ N, gcd(pr , m) = 1.

(4-31)

qlĂšn4.9ÂŒ ,d(n)´Ăˆ5Ÿê, l(4-31)ÂŒ d(n) = d(pr )d(m) = (r + 1)d(m).

(4-32)

ò(4-31)Ăš(4-32)“\(4-30)= m (r + 1)S(pr ) >3 . r p d(m) f (n) =

n d(n) ,

(4-33)

dž(4-33)ÂŒ ¤ (r + 1)S(pr ) > f (m). pr

(4-34)

u´, ŠâĂšn4.8ÚÚn4.10, l(4-34)ÂŒ : =kSmarandache ĂŞ12. ½n4.8y..

4.5 Smarandache 3n ĂŞiS ĂŠ?Âż ĂŞn, Ă?Âś Smarandache 3nĂŞiS an ½Ă‚Â? an = 13, 26, 39, 412, 515, 618, 721, 824, .... TĂŞ zÂ˜Â‡ĂŞĂ‘ÂŒ¹ŠÂ?ĂźĂœŠ, ÂŚ 12ĂœŠ´11ĂœŠ 3 . ~X, a12 = 1236, a20 = 2060, a41 = 41123, a333 = 333999, ....ĂšÂ˜ĂŞ ´Ă?ϐĂ˜;[Smarandache Ç 3 Š z[7]Ăš Š z[46]ÂĽ J Ă‘ , Ă“ ž ÂŚ ĂŻ Æ < ‚ ĂŻ Ă„ T ĂŞ 5 Â&#x;. ' u Ăš ˜ ÂŻK, ÂŽĂšĂĽĂ˜ ÆÜ 5Âż, Âż ˜ ĂŻĂ„¤J[47−49] .3Šz[49]ÂĽ, ĂœŠ+Ă&#x; ĂżSmarandache 3nĂŞiS ÂĽÂŒUvk ²Â?ĂŞ, Â?,Šz[49]ÂĽvk )ĂťTĂ&#x; ÂŽ, y² Âąe(Ă˜: (a) nÂ?Ăƒ²Â?Ă?fêž, an Ă˜ÂŒU´ ²Â?ĂŞ; 71

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

(b) nÂ? ²Â?êž, an Ă˜ÂŒU´ ²Â?ĂŞ; (c)XJan ´Â˜Â‡ ²Â?ĂŞ, @okn = 22a1 ¡32a2 ¡52a3 ¡112a4 ¡n1 , Ù¼(n1 , 330) = 1.

Ăš (Ă˜Â?Ă&#x;ÂŽ (5Jø ­Â‡Â?â, ÂŤ ĂŻĂ„TÂŻK ˜ g´ĂšÂ?{,

l,˜Â?ÂĄwÂŤĂ‘Smarandache 3nĂŞiS ˜ S35Â&#x;. 'uĂšÂ˜ĂŞ cn‘ ÂŚ Ú¯K´k¿Â , Â?Ă’´`´Ă„ 3a1 + a2 + ... + aN ˜‡(ƒ ÂŚĂšúª½ÜÏ Cúª? ²L{ßí ĂšOÂŽ, ÂŒÂąÂ‰Ă‘Â˜Â‡E, OÂŽúª, Ă™(J†N (ƒ/ÂŞ k', ´/ÂŞÂżĂ˜nÂŽ, Â…lÂĽĂ˜U ĂŒÂ‡ĂœŠ, =Ă˜U ĂŹCúª. u´, ! Äފ ln a1 + ln a2 + ... ln aN ĂŹC5ÂŻK, |^Ă? Â?{9 ĂŞ ? 5Â&#x;y² eÂĄ (Ă˜. ½ n 4.9 ĂŠ?¿¿ŠÂŒ ĂŞN, kĂŹCúª X ln an = 2N ¡ ln N + O(N). n6N

eÂĄy²T½n. Ă„k Ă„an ( , nÂ?k ê, =n = bk bk−1...b2 b1 ,Ù¼1 6 bk 6 9, 0 6 bi 6 9(i = 1, 2, ..., k − 1). u´dÂŚ{ ? {KÂŒ , 333...34 | {z } 6 n 6 |333...3 {z } k−1

ž, 3nÂ?k ê;

k

333...34 | {z } 6 n 6 333...33 | {z } k

k+1

ž, 3nÂ?k + 1 ê. dan ½Ă‚ĂĄÂ?

an = n ¡ (10k + 3), ½ an = n ¡ (10k+1 + 3). ĂŠ?¿¿ŠÂŒ ĂŞN, w, 3 ĂŞM , ÂŚ : 333...33 | {z } < n 6 333...33 | {z } . M

u´dcÂĄ ŠĂ›, kĂ° ÂŞ Y

16n6N

an =

3 Y

n=1

an ¡

33 Y

n=4

(4-35)

M+1

1 (10M −1) 3

an ¡ ¡ ¡

Y

n= 13 (10M−1 −1)+1 M−1

N!(10 + 3)3 ¡ (100 + 3)30 ¡ ... ¡ (10M + 3)3¡10 72

an ¡

N Y

an =

n= 13 (10M −1)+1 1

M

¡ (10M+1 + 3)N− 3 (10

−1)

. (4-36)

1oĂ™ SmarandacheS ĂŻĂ„

5Âż x → 0ž, k OÂŞln(1 + x) = x + O(x2 ),¤¹ M X

= =

k=1 M X k=1 M X k=1

k−1

ln(10k + 3)3¡10 k−1

3 ¡ 10

3 ¡ k ¡ ln 10 + k + O 10

k ¡ 10k−1 ¡ 3 ln 10 +

1 102k

9 M + O(1) 10

1 1 9 = M ¡ 10M ¡ ln 10 − (10M − 1) ¡ ln 10 + M + O(1) 3 27 10 1 = M ¡ 10M ¡ ln 10 + O(N). 3 M+1

N− 13 (10M −1)

(4-37)

1 M = N− 10 − 1 ln(10M+1 + 3) 3

ln(10 + 3) 1 M 10 − 1 (M + 1) ln 10 + O(1) = N− 3 1 = N ¡ M ¡ ln 10 − ¡ M ¡ 10M ¡ ln 10 + O(N). 3

(4-38)

A^EulerÂŚĂšúª½½ĂˆŠ 5Â&#x;, N´ X ln n = N ¡ ln N − N + O(1). ln(N!) =

(4-39)

16n6N

5Âż ÂŞ(4-35), Ă˜J Ă‘ OÂŞ 10M < N 6 10M+1 , ½Ü ln N = M ln 10 + O(1).

(4-40)

(ĂœĂ° ÂŞ(4-36)9ĂŹ?úª(4-37)-(4-40), ĂĄÂ? ĂŹCúª X ln an n6N

=

X

16n6N

ln n +

M X

k−1

ln(10k + 3)3¡10

1

M

+ ln(10M+1 + 3)N− 3 (10

−1)

k=1

= 2N ¡ ln N + O(N). u´½n4.9 y. w,, ڇÏCúªÂ„' oá, ´Ă„ 3Â?°( ĂŹCúª, „k –u?Â˜ĂšĂŻĂ„.

73

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

4.6 Smarandache kn ĂŞiS ĂŠu∀k ∈ N + , Ă?Âś Smarandache kn ĂŞiS {a (k, n)}½Ă‚Â?Ăš ĂŞ8, T8

ĂœÂĽ zÂ˜Â‡ĂŞĂ‘ÂŒ¹Š¤ßĂœŠ, ÂŚ 1 ĂœŠ´1Â˜ĂœŠ k . ~XĂžÂ˜!‰Ñ

Smarandache 3nĂŞiS {a (3, n)} = {13, 26, 39, 412, 515, 618, 721, 824, ...} , =Ă’´TĂŞ zÂ˜Â‡ĂŞĂ‘ÂŒ¹ŠÂ?ĂźĂœŠ, 1 ĂœŠ´1Â˜ĂœŠ 3 . ĂžÂ˜!¼‰Ñ 'u ln a (3, n) ފ5Â&#x;, y² ĂŹCúª X

n6N

ln a (3, n) = 2N ¡ ln N + O (N) .

'uĂšÂ˜ĂŞ ٌ5Â&#x;, –8q „vk<ĂŻĂ„, ïÄڇê 5Â&#x;´k¿Â , – ÂŒÂąÂ‡NĂ‘Ăš AĂ?ĂŞ A 9ŠĂ™5Â&#x;. !¤ 9 Ă? ĂŞĂ˜ ÂŁÂŒĂŤÂ„ Šz[6]. !|^Ă? 9|ĂœÂ?{ĂŻĂ„

n a (k, n) ފ5Â&#x;, ¿‰Ñ A‡k ĂŹCúª. ½ n 4.10 k ∈ N + , 2 6 k 6 5, KĂŠ?¿¿ŠÂŒ ĂŞx, kĂŹCúª X n 9 = ln x + O (x) . a (k, n) k ¡ 10 ¡ ln 10 16n6x

½ n 4.11 k ∈ N + , 6 6 k 6 9,K ¢êx ¿ŠÂŒÂž, EkĂŹCúª X n 9 = ln x + O (x) . a (k, n) k ¡ 10 ¡ ln 10 16n6x

AO/, k = 396 ž, d½n4.10„Œ¹íĂ‘eÂĄĂ­Ă˜: Ă­ Ă˜ 4.3 ĂŠ?¿¿ŠÂŒ ĂŞx, kĂŹCúª X n 3 = ln x + O (x) . a (3, n) 10 ¡ ln 10 16n6x

Ă­ Ă˜ 4.4 ĂŠ?¿¿ŠÂŒ ĂŞx, kĂŹCúª X n 3 = ln x + O (x) . a (6, n) 20 ¡ ln 10 16n6x

74

1oÙ SmarandacheS ïÄ

w,ùA ìCúªØ´ ©°(, ´Ä 3 °( ìCúªE,´ úm ¯ K, ò´UYïÄ 8I. e¡y²ùü ½n. y²½n4.10 ¥k = 293 ¹, aq/ ±íѽ n4.10 ¥k = 4, 59½n4.11. Äky²½n4.10¥k = 2. Ä a (2, n) ( , n ? L« k ê, =Ò´ n = bk bk−1 · · · b2 b1 , Ù¥ 1 6 bk 6 9, 0 6 bi 6 9, i = 1, 2, · · ·, k − 1. u´d¦{ ? {K , 10k−1 6 n 6 5 · 10k−1 − 1 , 2n k ê; 5 · 10k−1 6

n 6 10k − 1 , 2n k + 1 ê. da (2, n) ½Âá a (2, n) = n · 10k + 2 ,

½ö

a (2, n) = n · 10k+1 + 2 .

é?¿¿© êx, w,∃M ∈ N + ¦

5 · 10M 6 x < 5 · 10M+1 , u´kð ª X

16n6x 4 X

n a (2, n)

49 499 4999 X X X n n n n = + + + + a (2, n) a (2, n) a (2, n) a (2, n) n=1 n=5 n=50 n=500

···+ 4 X

M 5·10 X−1

n=5·10M−1

n + a (2, n)

X

5·10M 6n6x

n a (2, n)

49 499 4999 X X X 1 1 1 1 = + + + 2 3 4 10 + 2 n=5 10 + 2 n=50 10 + 2 n=500 10 + 2 n=1

···+

M 5·10 X−1

n=5·10M−1

1 M+1

10

+2

+

X

5·10M 6n6x

1 M+2

10

+2

5−1 50 − 5 500 − 50 5000 − 500 + + + + 2 10 + 2 10 + 2 103 + 2 104 + 2 5 · 10M − 5 · 10M−1 x − 5 · 10M ···+ +O 10M+1 + 2 10M+2 + 2 9 · 10 9 · 102 9 · 103 9 · 10M + O (1) = + + + · · · + 2 · 102 +2 2 · 103 +2 2 · 104 +2 2 · 10M+1 +2 =

75

(4-41)

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

9 ¡ 102 + 2 − 2 9 ¡ 103 + 2 − 2 9 104 + 2 − 2 + + + = 20 ¡ 102 + 2 20 ¡ 103 + 2 20 104 + 2 9 10M+1 + 2 − 2 + O (1) ¡¡¡+ 20 10M+1 + 2 ! M X 9 1 = M +O + O (1) 20 10m m=1

=

9 M + O (1) . 20

(4-42)

5Âż ÂŞ(4-41) , ĂŠĂŞ

k OÂŞ

M ln 10 + ln 5 6 x < ln 5 + (M + 1) ln 10 ½Ü M= ¤¹dª( 4-42) ÏCúª X 16n6x

1 ln x + O (1) . ln 10

n 9 = ln x + O (x) . a (k, n) 20 ¡ ln 10

u´y² ½n4.10ÂĽk = 2 Âœš. y3y²½n4.10 ÂĽk = 3 Âœš. Ă„ ĂŞ a (3, n) ( , n ›?›LÂŤ Â?k ê, =Ă’´ n = bk bk−1 ¡ ¡ ¡ b2 b1 , Ù¼ 1 6 bk 6 9, 0 6 bi 6 9, i = 1, 2, ¡ ¡ ¡k − 1. u´dÂŚ{ ? {KÂŒ , ¡ ¡ ¡ 34} 6 n 6 333 ¡ ¡ ¡ 33} |333 {z | {z k−1

ž, 3nÂ?k ê;

k

333 ¡ ¡ ¡ 34} 6 n 6 333 ¡ ¡ ¡ 33} | {z | {z k

k+1

ž, 3nÂ?k + 1 ê. u´ nÂ?k êž, da (3, n) ½Ă‚

½Ü

a (3, n) = n ¡ 10k + 3 , a (3, n) = n ¡ 10k+1 + 3 . 76

1oĂ™ SmarandacheS ĂŻĂ„

ĂŠ?¿¿ŠÂŒ ĂŞx, w,∃M ∈ N + , ÂŚ 333 ¡ ¡ ¡ 33} 6 x < |333 {z ¡ ¡ ¡ 33}, | {z M

u´kð ª X 16n6N 3 X

(4-43)

M+1

n a(3, n)

33 333 3333 X X X n n n n = + + + + a(3, n) n=4 a(3, n) a(3, n) n=334 a(3, n) n=1

¡¡¡ =

X

n= 13 ¡(10M−1 −1)+1

3 X

n + a(3, n)

X

1 (10M −1)+16N6x 3

n a(3, n)

33 333 3333 X X X 1 1 1 1 + + + + 2+3 3+3 4+3 10 + 3 10 10 10 n=1 n=4 n=34 n=334

¡¡¡ + =

n=34)

1 ¡(10M −1) 3

1 ¡(10M −1) 3

X

1

n= 13 ¡(10M−1 −1)+1

10M

+3

X

+

1 ¡(10M −1)+16n6x 3

1 10M+1

+3

3 33 − 3 333 − 33 3333 − 333 + 2 + + + 10 + 3 10 + 3 103 + 3 104 + 3 ! 1 1 1 M M−1 M ¡ (10 − 1) − (10 − 1) x − ¡ (10 − 1) 3 3 ¡¡¡ + 3 +O 10M + 3 10M+1 + 3

3¡1 3 ¡ 10 3 ¡ 102 3 ¡ 103 + 2 + 3 + 4 + 10 + 3 10 + 3 10 + 3 10 + 3 3 ¡ 10M−1 + O(1) ¡¡¡ + 10M + 3 3 ¡ (10 + 3 − 3) 3 ¡ (102 + 3 − 3) 3 ¡ (103 + 3 − 3) 3 ¡ (104 + 3 − 3) = + + + + 10 ¡ (10 + 3) 10 ¡ (102 + 3) 10 ¡ (103 + 3) 10 ¡ (104 + 3) 3 ¡ (10M+1 + 3 − 3) ¡¡¡ + + O(1) 10 ¡ (10M + 3) ! M X 3 1 = M +O + O(1) 10 10m m=1

=

=

3 M + O(1). 10

(4-44)

5¿ ª(4-43) =´

u´k Oª

1 1 ¡ 10M − 1 6 x < ¡ 10M+1 − 1 , 3 3

M=

1 ln x + O (1) . ln 10 77

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

¤¹dª(4-44) ÏCúª X

16n6x

3 n = ln x + O (x) . a (3, n) 10 ¡ ln 10

u´ ¤ ½n4.10 ÂĽk = 3 y². |^Ă“ Â?{Â?ÂŒ¹íĂ‘½n4.10 ÂĽk = 4, 5 (Ă˜. –u½n4.11 y², Ăš½n4.10 y²aq, Â?´3ŒÚ¼Ên Š{Ă˜Ă“, Ăš pĂ’Ă˜Â˜Â˜Ăž~.

4.7 Smarandache ²Â?ĂŞ ĂŠ?¿šK ĂŞn, ^SP (n)LÂŤn Smarandache ²Â?ĂŞ, =Ă’´ÂŒu½ un  ²Â?ĂŞ. ~XTĂŞ cA‘Â?: 0, 1, 4, 4, 4, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 25, ¡ ¡ ¡ . ^IP (n)LÂŤn Smarandache ÂŒ²Â?ĂŞ, =Ă’´Ă˜Â‡Ln  ÂŒ ²Â?ĂŞ.ڇê cA‘Â?: 0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, ¡ ¡ ¡ . Sn =

(SP (1) + SP (2) + ¡ ¡ ¡ + SP (n)) ; n

In =

(IP (1) + IP (2) + ¡ ¡ ¡ + IP (n)) ; n

Kn = Ln =

p n

SP (1) + SP (2) + ¡ ¡ ¡ + SP (n);

p n

IP (1) + IP (2) + ¡ ¡ ¡ + IP (n).

3Šz[7]ÂĽ, {7ÛêZĂŚĂ?ϐĂ˜;[Smarandache ÇJĂ‘ Ú߇ê , ¿ïÆ<╀§ ÂˆÂŤ5Â&#x;, k'Ăš SNĂšk' ¾Í Šz[51, 52, 57].3Š z[56]ÂĽ, F Kenichiro KashiharaƏ2gÊÚ߇ê ) , , ӞJĂ‘ ĂŻĂ„ n 4 SInn !Sn − In ! K Ln 9Kn − Ln ùÑ5ÂŻK, XJĂ‚Ăą, ¿ŒĂ™4 . 3Šz[60]ÂĽ,

ÂƒĂ„gĂŻĂ„ ĂšA‡ÞŠ ĂŹC5ÂŻK, Âż|^Ă? 9)Ă›Â?{y² eÂĄA‡( Ă˜:

78

1oÙ SmarandacheS ïÄ

½ n 4.12 é?¿¢êx > 2, kìCúª X 3 x2 SP (n) = + O(x 2 ); 2 n6x

X

IP (n) =

n6x

3 x2 + O(x 2 ). 2

dd½ná íÑe¡ íØ: í Ø 4.5 é?¿ ên, kìCúª 1 Sn = 1 + O(n− 2 ); In

94 ª lim

n→∞

Sn = 1. In

í Ø 4.6 é?¿ ên, kìCúª 1 Kn = 1 + O(n− 2 ); Ln

94 ª Kn = 1, n→∞ Ln lim

lim (Kn − Ln ) = 0.

n→∞

, , 'uSn − In ìC5¯K, 8q vk<ïÄ, vk3yk ©

z ¥ w ., , ö @ ù ¯ K ´ k , Ù Ï 3 u ¡ § ) û ± é ©z[56]¥ ¯K ± £ , xþ ÷ éÒ;, ¡ ± xÑü«ê SP (n)9IP (n) «O. ÖÄu©z[60]¥ g ¿(ÜÓa Ü¿±9Ø °(?n, ïÄ Sn − In ìC5¯K, ¼ r ìCúª, äN/` Ò´y² e¡ ½n.

½ n 4.13 éu?¿ ên > 2, kìCúª Sn − In =

4√ n + O(1). 3

ù (J Ö ©z[60] Øv, Ó ò©z[56]¥éê Sn9In JÑ ¤k¯K )û. ,, dT½n ±íÑe¡ 4 : 1

lim (Sn − In ) n = 1;

n→∞

lim

Sn − In 4 √ = . 3 n 79

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

eÂĄy²d½n. |^Ă? Â?{9EulerÂŚĂšúªŠOĂŠSn9In ?1š~°( O,  ª|^߇°( O‰Ñ½n y².ĂŠ?Âż ĂŞn > 2, w, 3Â?˜ 1

ĂŞM áv:M 2 < n 6 (M + 1)2 , =M = n 2 + O(1).u´k 1X 1 X 1 X S(n) = SP (k) = SP (k) + SP (k) n n n 2 2 k6n k6M M <k6n X 1 X 1 X SP (k) + (M + 1)2 = n n 2 2 2 h6M (h−1) <k6h

M <k6n

1 X 2 1 = (h − (h − 1)2 )h2 + (n − M 2 )(M + 1)2 n n h6M 1 X 1 = (2h3 − h2 ) + (n − M 2 )(M + 1)2 n n h6M 2

M (M + 1)2 M (M + 1)(2M + 1) 1 − + (n − M 2 )(M + 1)2 2n 6n n 2 M (M + 1)(2M + 1) M (M + 1)2 2 − . = (M + 1) − 6n 2n =

(4-45)

Ă“nŠâIP (n) ½Ă‚, Â?kOÂŽúª 1X 1 X 1 X In = IP (k) = IP (k) + IP (k) n n n 2 2 k6n k<M M 6k6n X 1 X 1 X = IP (k) + M2 n n 2 2 2 h6M (h−1) 6k<h

M 6k6n

1 X 2 1 = (h − (h − 1)2 )(h − 1)2 + (n − M 2 + 1)M 2 n n h6M 1 X 1 = (2h3 − 5h2 + 4h − 1) + (n − M 2 + 1)M 2 n n h6M 2

M (M + 1)2 5M (M + 1)(2M + 1) 2M (M + 1) M (n − M 2 + 1)M 2 − + − + 2n 6n n n n 2 2 2 M (M − 2M − 3) 5M (M + 1)(2M + 1) 2M + M = M2 − − + . (4-46) 2n 6n n =

u´d(4-45)Ăš(4-46) M (M + 1)(2M + 1) M 2 (M + 1)2 Sn − In = (M + 1)2 − − − 6n 2n M 2 (M 2 − 2M − 3) 5M (M + 1)(2M + 1) 2M 2 + M 2 M − − + 2n 6n n 3 2M + 7M = 2M + 1 − . (4-47) 3n 5Âż 1

M = n 2 + O(1), 80

1oĂ™ SmarandacheS ĂŻĂ„

d(4-47)ÂŞBÂŒĂ­Ă‘ Sn − In = 2M − =

2M 4 + O(1) = M + O(1) 3 3

4√ n + O(1). 3

81

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

1ĂŠĂ™ Ă™§êĂ˜ÂŻK 5.1 škFibonacciê†LucasĂŞ Ă° ÂŞ

S

¯¤¹ , FibonacciS {Fn } †LucasS {Ln}(n = 0, 1, 2, ...) ´d ‚548 Fn+2 = Fn+1 + Fn , Ln+2 = Ln+1 + Ln

5½Ă‚ , Ù¼n > 0, F0 = 0, F1 = 1, L0 = 2 Â…L1 = 1. Ú߇S 3êÆ!ĂŻĂ“ Æ!OÂŽĂ…D–)ÔÆ ˆ‡+Â?Ă‘užX4Ă™­Â‡ Š^. Ă?d'uFn ĂšLn ˆ ÂŤÂˆ 5Â&#x; ĂŻĂ„ĂĄĂš IS ¯þÆÜ 81. R.L.Duncan[76] ĂšL.Kuipers[78] ŠO 31967cĂš1969cy² log Fn ´ 1 ˜—ŠĂ™ . Neville Robbins[81] 31988cĂŻĂ„

/Xpx2 Âą 1, px3 Âą 1 (Ù¼p ´ÂƒĂŞ) FibonacciĂŞ. 'uFibonacciĂŞ Ă° ÂŞĂšÂ?ÂĄ SN, Â?kĂ˜ ʤ

(J. ~X, m X

Fn = Fm+2 − 1,

n=0 m−1 X

F2n+1 = F2m ,

n=0

m X n=0

m X

nFn = mFm+2 − Fm+3 + 2, n m−n Cm Fc−n Fkn Fk+1 = Fmk+c ,

n=0

n Ù¼Cm =

m! n!(m−n)! .

2004c, ĂœŠ+[84] Â?šFn ĂšLn A‡Â?2Â? Ă° ÂŞ X

Fm(a1 +1) Fm(a2 +1) . . . Fm(ak+1 +1) =

a1 +a2 +¡¡¡+ak+1 =n

X

F k+1 (k) (−1)mn m U 2k k! n+k

Lma1 Lma2 . . . Lmak+1

a1 +a2 +¡¡¡+ak+1 =n+k+1

82

im Lm , 2

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

m(n+k+1)

= (−1)

k+1 m+2

2 X k! h=0

i

2

Lm

h

(k + 1)! (k) Un+2k+1−h h!(k + 1 − h)!

im Lm , 2

Ù¼k, m ´?Âż ĂŞ, n, a1 , a2 , . . . , ak+1 ´ÂšK ĂŞÂ…i ´âˆ’1 ²Â?Š, Uk (x) ´1 aChebyshevþ‘ª. ´w[90] 3dĂ„:Ăžq 'uFibonacciþ‘ª A‡ð ÂŞ n

X

a1 +a2 +¡¡¡+ak =n

Fa1 +1 (x) ¡ Fa2 +1 (x) ¡ ¡ ¡ Fak +1 (x) =

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ¡ xn−2m ,

Ù¼Fibonacciþ‘ªFn(x)d ‚548úªFn+2(x) = xFn+1 (x) + Fn (x)½Ă‚, Â… n Ă?ŠÂ?F0 (x) = 0,F1 (x) = 1; Cm =

m! n!(m−n)! ,[z]LÂŤĂ˜Â‡Lz  ÂŒ ĂŞ;

x = 1ž,

Fn (x) = Fn , dd A‡'uFibonacciĂŞ Ă° ÂŞ,ÂąeA‡úªĂ‘ĂŠ?Âż ĂŞk Ăšn¤å n

X

a1 +a2 +¡¡¡+ak =n+k

Fa1 ¡ Fa2 ¡ ¡ ¡ Fak =

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ,

n

X

a1 +a2 +¡¡¡+ak =n+k

F2a1 ¡ F2a2 ¡ ¡ ¡ F2ak = 3k ¡ 5

n−k 2

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ¡ 5−m ,

n

X

a1 +a2 +¡¡¡+ak =n+k

F3a1 ¡ F3a2 ¡ ¡ ¡ F3ak = 22n+k

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ¡ 16−m ,

n

X

a1 +a2 +¡¡¡+ak =n+k

F4a1 ¡ F4a2 ¡ ¡ ¡ F4ak = 3n ¡ 7k ¡ 5

n−k 2

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ¡ 45−m ,

n

X

a1 +a2 +¡¡¡+ak =n+k

F5a1 ¡ F5a2 ¡ ¡ ¡ F5ak = 5k ¡ 11n

[2] X

m=0

m k−1 Cn+k−1−m ¡ Cn+k−1−2m ¡ 121−m .

!ĂŒÂ‡Ă?LrxnL¤Chebyshevþ‘ª5 Â?šFibonacciê†LucasĂŞ o ‡ð ÂŞ. Â?Ă’´, ‡y²eÂĄ ߇½n. ½ n 5.1 ĂŠ?¿šK ĂŞn Ăš ĂŞm, kĂ° ÂŞ L2n m

=

(2n)! (−1)mn (n!)2

L2n+1 m

= (2n + 1)!

+ (2n)!

n X k=0

n X k=1

(−1)m(n−k) L2km ; (n − k)!(n + k)!

(−1)m(n−k) L(2k+1)m . (n − k)!(n + k + 1)! 83

Smarandache ¼ê9Ù '¯KïÄ

½ n 5.2 é?¿ K ên Ú êm, kð ª n

L2n m

(2n)! X (−1)m(n−k) (2k + 1) F(2k+1)m ; = Fm (n − k)!(n + k + 1)! k=0

n

L2n+1 m

2(2n + 1)! X (−1)m(n−k) (k + 1) = F2(k+1)m . Fm (n − k)!(n + k + 2)! k=0

!ò Ñ3½ny²¥I ^ A Ún. 3 ÑÚn cÄkI 1 aChebyshevõ ªTn(x) Ú1 aChebyshevõ ªUn(x) (n = 0, 1, · · · ) L

ª

n n i p p 1 h x + x2 − 1 + x − x2 − 1 , 2 n+1 n+1 p p 1 Un (x) = √ x + x2 − 1 − x − x2 − 1 . 2 x2 − 1

Tn (x) =

,§ ±^Xe48úª5½Â:

Tn+2 (x) = 2xTn+1 (x) − Tn (x), Un+2 (x) = 2xUn+1 (x) − Un (x), Ù¥n > 0, T0 (x) = 1, T1 (x) = x, U0 (x) = 1 U1 (x) = 2x. e¡ ÑXeA Ún. Ú n 5.1 é?Û êm Ún, kð ª Tn (Tm (x)) = Tmn (x), Um(n+1)−1 (x) Un (Tm (x)) = . Um−1 (x) y²: ë ©z[84]. Ú n 5.2 i ´−1 ² , m Ún ´?¿ ê, Kkð ª i Un = in Fn+1 , 2 i in = Ln , Tn 2 2 i imn Tn Tm = Lmn , 2 2 Fm(n+1) i Un Tm = imn . 2 Fm n y²: w,kUn 2i = in Fn+1 , Tn 2i = i2 Ln, K âÚn5.1, qN´/ {e

ü úª. ùÒy² Ún5.2.

84

1ÊÙ Ù§êدK

Ú n 5.3 é?¿ K ên, ∞

X 1 x ≡ an·0 T0 (x) + an·k Tk (x) 2 n

(5-1)

k=1

xn ≡

∞ X

bn·k Uk (x),

(5-2)

k=0

Kk

an·k =

bn·k =

 

2n! (n−k)!!(n+k)!! ,

n > k, n + k ´óê; Ù¦.

 0,

 

2(k+1)n! , (n−k)!!(n+k+2)!!

n > k, n + k ´óê; Ù¦.

 0,

y²: , Chebyshevõ ªkéõ5 (ë ©z[92]). ~X   m 6= n,  Z 1  0, Tm (x)Tn (x) √ dx = π2 , m = n > 0, 2  1−x −1   π, m = n = 0; Tn (cos θ) = cos nθ;

   Z 1p  0, 1 − x2 Um (x)Un (x) dx = π2 ,  −1   π,

Un (cos θ) =

m 6= n,

m = n > 0,

(5-3)

(5-4)

(5-5)

(5-6)

(5-7)

m = n = 0;

sin(n + 1)θ . sin θ

(5-8)

(x) Äky²(5-3)ª. é?¿ K êm, Äk (5-1)ªü>Ó ¦± √Tm , 2ü> 1−x2

l−1 1 È©, A^(5-5)ª Z 1 n Z 1 Z 1 ∞ X x Tm (x) 1 Tm (x)T0 (x) Tm (x)Tk (x) √ √ √ dx = an·0 dx + an·k dx 2 2 1 − x2 1 − x2 1 − x −1 −1 −1 k=1 π = an·m , (m = 0, 1, 2, · · · ) 2 Ïd, an·m

2 = π

Z

1

−1

xn Tm (x) √ dx. 1 − x2

85

Smarandache ¼ê9Ù '¯KïÄ

x = cos t, â(5-6)ªk Z 2 π an·m = cosn t cos mt dt π 0 Z 2 π = cosn t (cos(m − 1)t cos t − sin(m − 1)t sin t) dt π 0 Z Z 2 π 2 π n+1 = cos t cos(m − 1)t dt − cosn t sin(m − 1)t sin t dt π 0 π 0 Z π 2 1 sin(m − 1)t d(cosn+1 t) = an+1·m−1 + · π n+1 0

π

2 1 n+1 = an+1·m−1 + · · cos t · sin(m − 1)t

π n+1 0 Z 2 m−1 π n+1 − · cos t cos(m − 1)t dt π n+1 0 m−1 = an+1·m−1 − · an+1·m−1 n+1 n−m+2 = · an+1·m−1 n+1 n−m+2 n−m+4 = · · an+2·m−2 n+1 n+2 =··· n−m+2 n−m+4 n − m + 2m = · ··· · an+m·0 n+1 n+2 n+m =

(n+m)!! (n−m)!! (n+m)! n!

· an+m·0 .

e¡O an+m·0 , w, an+m·0 {z, In = ê, K

Rπ 0

2 = π

Z

π

cosn+m t dt.

0

cosn t dt. n Ûê , Ï f (x) = cos x 3«m[0, π] þ´Û¼

In =

Z

π

Z

π

cosn t dt = 0;

(5-9)

0

n óê , k Z

π

cos t dt = cosn−1 t d sin t 0

0 Z π

n−1 π = sin t cos t 0 + (n − 1) sin2 t · cosn−2 t dt 0 Z π = (n − 1) 1 − cos2 t cosn−2 t dt Z0 π Z π n−2 cos t dt − (n − 1) cosn t dt = (n − 1)

In =

n

0

0

86

1ÊÙ Ù§êدK

= (n − 1)In−2 + (n − 1)In .

(5-10)

Ïd 'uIn 4íúª In =

n−1 · In−2 . n

(5-11)

Kdª(5-11) n−1 n =··· n−1 = n

In =

dIn ½Â I0 =

Z

· In−2 = ·

n−1 n−3 · · In−4 n n−2

n−3 1 · · · · I0 , n−2 2

π

0

cos t dt =

0

Z

0

(5-12)

π

1 · dt = π.

(5-13)

Ïd(ܪ(5-9), (5-12)Ú(5-13), Z π In = cosn t dt 0  (n−1)!! π, n ´óê, n!! =  0, n ´Ûê. dd

Z 2 π an+m·0 = cosn+m t dt π  0  2(n+m−1)!! , n + m ´óê, (n+m)!! =  0, n + m ´Ûê.

Ïd an·m =

 (n+m)!!  (n−m)!! (n+m)! n!

2(n+m−1)!! (n+m)!!

=

2n! , (n−m)!!(n+m)!!

n > m, n + m ´óê, Ù¦.

0,

-m = k, á= (5-3)ª. √

^ a q {, 5 y ²(5-4)ª. é ? ¿ K êm, Ä k (5-2)ª ü > Ó ¦

± 1 − x2 Um (x), 2l−1 1 È©, |^(5-7)ª Z 1p Z 1p ∞ X n 2 1 − x x Um (x) dx = bn·k 1 − x2 Um (x)Uk (x) dx −1

k=0

π = bn·m , 2

87

−1

(m = 0, 1, 2, · · · )

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

-m = k, Kk bn¡k

2 = π

Z

1 −1

p

1 − x2 xn Uk (x) dx.

x = cos t Šâ(5-8)ÂŞ, k Z 2 Ď€ bn¡k = cosn t ¡ sin(k + 1)t ¡ sin t dt Ď€ 0 Z Ď€ 2 −1 = ¡ sin(k + 1)t d(cosn+1 t) Ď€ n+1 0 Z Ď€

Ď€ 2 −1 n+1 n+1

= ¡ cos t ¡ sin(k + 1)t 0 − (k + 1) cos t ¡ cos(k + 1)t dt Ď€ n+1 0 Z 2 k+1 Ď€ = ¡ cosn+1 t ¡ cos(k + 1)t dt Ď€ n+1 0 k+1 = ¡ an+1¡k+1 . n+1 |^(5-3)ÂŞ, k bn¡k =

 

2(k+1)n! , (n−k)!!(n+k+2)!!

n > k, n + k ´óê; Ă„K.

 0,

Ă?d (5-4)ÂŞ. ÚÒy² Ăšn2.3. Ăš n 5.4 ĂŠ?¿šK ĂŞn, rxn L¤Xe /ÂŞ n

x

2n

(2n)! 2(2n)! X 1 = n T (x) + T2k (x) 0 2 n 4 (n!) 4 (n − k)!(n + k)! k=1

=

n (2n)! X

4n

k=0

2k + 1 U2k (x), (n − k)!(n + k + 1)! n

x

2n+1

(2n + 1)! X 1 = T2k+1 (x) n 4 (n − k)!(n + k + 1)! k=0

=

n (2n + 1)! X

4n

k=0

k+1 U2k+1 (x). (n − k)!(n + k + 2)!

y²: 3Ăšn5.3 ÂŞ(5-3)Úª(5-4)ÂĽ, ank 6= 0 ž, n + k 7Â?óê, =n †k ä

kÂƒĂ“ Ûó5, |^ĂšÂ˜:, òy²Ăšn5.4.

Ă„ky²1˜‡ ÂŞ, Šâª(5-1)9(5-3), k ∞

X 1 x2n = a2n¡0 T0 (x) + a2n¡k Tk (x) 2 k=1

1 = a2n¡0 T0 (x) + 2

n X

a2n¡2k T2k (x)

k=1

88

1ÊÙ Ù§êدK n

X 2(2n)! 2(2n)! 1 = · T0 (x) + · T2k (x) 2 (2n)!!(2n)!! (2n − 2k)!!(2n + 2k)!! k=1

=

(2n)! T0 (x) + 4n (n!)2

n 2(2n)! X

4n

k=1

1 · T2k (x), (n − k)!(n + k)!

, ¡, dª(5-2)Úª(5-4) x

2n

= = =

∞ X

k=0 n X k=0 n X k=0

=

b2n·k Uk (x) b2n·2k U2k (x) 2(2k + 1)(2n)! · U2k (x) (2n − 2k)!!(2n + 2k + 2)!! n

(2n)! X 2k + 1 · U2k (x). n 4 (n − k)!(n + k + 1)! k=0

é u 1 ª, Ï 2n + 1 ´ Û ê, ¤ ±a2n+1·0 = 0, Ï d d Ú n5.3¥ ª(5-1)Ú ª(5-3), k x

2n+1

=

∞ X

a2n+1·k Tk (x)

k=1

=

n X

a2n+1·2k+1 T2k+1 (x)

k=0

=

n X k=0

2(2n + 1)! · T2k+1 (x) (2n − 2k)!!(2n + 2k + 2)!! n

(2n + 1)! X 1 · T2k+1 (x), = n 4 (n − k)!(n + k + 1)! k=0

, ¡, dª(5-2)Úª(5-4) x2n+1 =

∞ X

b2n+1·k Uk (x)

k=0

=

n X

b2n+1·2k+1U2k+1 (x)

k=0

=

n X k=0

2(2k + 2)(2n + 1)! · U2k+1 (x) (2n − 2k)!!(2n + 2k + 4)!! n

(2n + 1)! X k+1 = · U2k (x). n 4 (n − k)!(n + k + 2)! k=0

ùÒy² Ún5.4.

89

Smarandache ¼ê9Ù '¯KïÄ

y35 Ñ1 !ü ½n y². ½n5.1 y²: âÚn5.4, -x = Tm (x), Kk n

2n Tm (x)

1 (2n)! 2(2n)! X T2k (Tm (x)), = n T (T (x)) + 0 m 2 n 4 (n!) 4 (n − k)!(n + k)! k=1

n

2n+1 Tm (x)

(2n + 1)! X 1 = T2k+1 (Tm (x)), n 4 (n − k)!(n + k + 1)! k=0

2-x =

i 2

¿ \þª, âÚn5.2 ± n

i2mk i2mn 2n (2n)! 2(2n)! X 1 L = + L2mk , 22n m 4n (n!)2 4n (n − k)!(n + k)! 2 k=1

n X

i(2n+1)m 2n+1 (2n + 1)! L = 22n+1 m 22n

k=0

= L2n m

=

(2n)! (−1)mn (n!)2

L2n+1 = (2n + 1)! m

1 i(2k+1)m Lm(2k+1) . (n − k)!(n + k + 1)! 2

+ (2n)!

n X k=1

n X k=0

(−1)m(k−n) L2km , (n − k)!(n + k)!

(−1)m(k−n) Lm(2k+1) . (n − k)!(n + k + 1)!

ùÒ ¤ ½n5.1 y². ½n5.2 y²: |^Ún5.4, k n

2n (x) Tm

(2n)! X 2k + 1 = n U2k (Tm (x)), 4 (n − k)!(n + k + 1)! k=0

n

2n+1 Tm (x)

(2n + 1)! X k+1 = U2k+1 (Tm (x)), n 4 (n − k)!(n + k + 2)! k=0

¿ -x = 2i , âÚn5.2 k

n

Fm(2k+1) i2mn 2n (2n)! X 2k + 1 Lm = n i2mk , 2n 2 4 (n − k)!(n + k + 1)! Fm k=0

n

Fm(2k+2) im(2n+1) 2n+1 (2n + 1)! X k+1 Lm = i(2k+1)m . 2n+1 n 2 4 (n − k)!(n + k + 2)! Fm k=0

z{ L2n m

n X (−1)m(k−n) (2k + 1) Fm(2k+1) = (2n)! , (n − k)!(n + k + 1)! Fm k=0

L2n+1 m

n X (−1)m(k−n) (k + 1) F2m(k+1) = 2(2n + 1)! . (n − k)!(n + k + 2)! Fm k=0

90

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

ÚÒy² ½n5.2.

5.2 Dirichlet L-Ÿê†n Ăš ĂŠÂˆÂŤn Ăš O´)Ă›ĂŞĂ˜¼ ­Â‡ ïđKƒ˜. p ´ÂƒĂŞ, f (x) = a0 + a1 x + ¡ ¡ ¡ + ak xk ´Â˜Â‡ ĂŞXĂŞ kgþ‘ª, Â…áv(p, a0 , a1 , . . . , ak ) = 1. @

31932c, L. J. Mordell[93] ĂŻĂ„ Xe¤ã ƒCĂŞn Ú¿ Ă?Âś ½n p−1 X 1 f (x) e ≪ p1− k , p x=1 Ù¼e(y) = e2Ď€iy . 3›c

uێ[94,95,101] ĂšDnä[102] òÚÂ˜ÂŻK*Ă? ߇CĂž, = p−1 X p−1 X 2 f (x, y) e ≪ p2− k , p x=1 y=1

Ù¼f (x, y) ´'u߇CĂžx Ăšy kgþ‘ª, ´¢Ăƒ ´, ĂšÂ˜(JĂ˜UÚßCĂž n Ăšpƒ=†. ƒ

, L. Carlitz ĂšS. Uchiyama[103] qU? Šz[93] (J

p−1

X f (x) √

e

6 k p,

p x=1

Ù¼k > 2.

!ÂĽ, ½Ă‚Xe‘A n Ăš q X

χ(a)e

a=1

f (a) , q

Ù¼χ ´Â˜Â‡ q DirichletA , Â…q ∤ (a0 , a1 , . . . , ak ). f (a) = naž, Tn ĂšC¤ GaussĂš G(n, χ) =

q X

χ(a)e

a=1

AO/, n = 1, P

χ = χ0 ž, P

an q

q X

a τ (χ) = χ(a)e . q a=1

Cq (n) =

q X

a=1

91

′

na e q

,

.

Smarandache ¼ê9Ù '¯KïÄ

¡ RamanujanÚ, Ù¥

q X

′

L«é¤k qp a ¦Ú. ù´þ¡¤?Ø n Ú

a=1

AÏ ¹(eq = p ´ ê f (a) = na), ,§ 5 ) éõ. 'uGaussÚ 5 , w,k G(n1 , χ) = G(n2 , χ),

n1 ≡ n2 (modq),

G(−n, χ) = χ(−1)G(n, χ), G(n, χ) = χ(n)τ ¯ (χ),

G(n, χ) ¯ = χ(−1)G(n, χ), ¯

(n, q) = 1,

Cq (n) = Cq (1) = τ (χ0 ),

(n, q) = 1,

d , χ ´ q A , χ0 ´ÌA , q, n ´ ê. , , eχ, χ1 Úχ2 ©O q, q1 Úq2 A , ÷v^ q = q1 q2 , (q1 , q2 ) = 1, χ = χ1 χ2 , KGaussÚ ±©) eª G(n, χ) = χ1 (q2 )χ2 (q1 )G(n, χ1 )G(n, χ2 ). eχ q A (' u A ½ Â ë © z[6, 106,]), χ(n)( q)´ d A χ∗ ( q ∗ ) Ñ , q1 ´ Úq ∗ k Ó Ï f(Ø O ­ ê) q Ø ê, K é(n.q) > 1, q ∗ 6= (n,q1q1 ) , kG(n, χ) = 0; q ∗ = (n,q1q1 ) , k n q q q ∗ ∗ −1 G(n, χ) = χ¯ χ µ ∗ φ(q)φ τ (χ∗ ), (n, q) q ∗ (n, q) q (n, q) (n, q) Ù¥, µ(n) ´M¨obius¼ê, φ(q) ´Euler¼ê. 'uτ (χ) ­ 5 Ò´eχ ´ q(q ?¿ ê) A , √ K|τ (χ)| = q, é?¿ χ q, kXeØ ª(ë ©z[6] Ú[106]) |τ (χ)| 6

√

q.

, , τ (χ) k ©)ª. eχq ⇔ χ∗q∗ , k q q ∗ τ (χ) = χ µ ∗ τ (χ∗ ). ∗ q q ´, χ ´ DirichletA , f (a) ´ kgõ ª , p X f (a) χ(a)e p a=1 Cz´4ÙØ5Æ . kNõÆöé§ ½ ïÄ. é êq > 3, k õ ªf (x) gê, φ(q) Euler¼ê, χ ´ q Dirichlet A , 92

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

e(k, qφ(q)) = 1, Mó¸[108] ‰Ñ ڍ‘A n Ăš ˜‡ ÂŞ

q 4 q Y X X

X f (a)

Îą+2 2 2 χ(a)e Îą+1− ,

= φ (q)q

q p ι m=1 χ mod q a=1

Ù¼

Q

pÎą kq

p kq

LÊ¤kávpÎą |q Â…pÎą+1 ∤ q p ÂŚĂˆ.

q, m, n 9k ´ ĂŞÂ…ávq, k > 1, (q, k) = 1ž, 4uw[109] ĂŻĂ„ Xe/ÂŞ ¡

ĂœÂ?ĂŞĂš C(m, n, k, χ; q) =

q X

χ(a)e

a=1

eÂĄ ÂŞ q X X

χ mod q m=1

4

2 2

|C(m, n, k, χ; q)| = q φ (q)

Y

pÎą kq Îą>2

mak + na q

,

Îą + 1 + 2(k, p − 1) Îą+1− p

¡

Y 2((k, p − 1) + 1) 1 (k, p − 1)2 − 2+ ¡ 2− , p p p(p − 1) pkq

Ù¼

Q

pÎą kq ½Ă‚Ă“Ăž.

4ĂŻĂŚ!½­2ĂšĂ?7[110] Â?´òXe/ÂŞ ƒCĂŞn Úí2 3 ÂŤmĂž ¿‰Ñ OÂŞ X

x<m6x+y

1

Λ(m)e(mk ι) ≪ (qx)Ǎ

1

q 2 yÎť 2 1

x2

5

1 2

1 2

1 6

1 2

+q x Îť +y x

3 10

+

x4

1

Îť6

+

x 1

1

q2Îť2

!

,

Ù¼, Λ(n) ´von MangoldtŸê, k > 1 ´ ĂŞ, x, y Ă‘Â?¢êÂ…2 6 x 6 y, Îą = a q

+ d,1 6 a 6 q, (a, q) = 1, Îť = |d|xk + x2 y −2 . Ă™§'un Ăš SNžÍ Š

z[111-119].

!´òn Ăš

p X a=1

χ(a)e

f (a) , p

†Dirichlet L-Ÿê?1\ , ‰ÑŒ‚ \ ފ ĂŹCúª. 'uDirichelet L-Ÿê \ ފ, IS ¯þÆÜÑʧ?1 ĂŻĂ„(ĂŤ Šz[120-127]). 32002c´wĂšĂœŠ+[128] ҉Ñ Dirichlet L-ŸêÚτ (χ) \ ފ ĂŹCúª. X |Ď„ (χ)|m |L(1, χ)|2k

χ6=χ0

k−1 Y 1 − C2k−2 1 2k−1 Y 2 2k−1 =N φ (N)Îś (2) 1− 2 1− p p2 p|q p∤q m +O q 2 +ÇŤ , m −1 2

93

!

Y

p|M

m m p 2 +1 − 2p 2 + 1

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ù¼q > 3 Â? ĂŞÂ…ávq = M N, (M, N) = 1, M =

Q

pkq n Â…p2 ∤ q p ÂŚĂˆ), Cm =

m! . n!(m−n)!

p(

Q

pkq

LÊ¤kávp|q

u´, !rĂš ˜‡2Ă‚n چDirichlet L-Ÿê(Ăœ3Â˜ĂĽ, ĂŻĂ„§Â‚\ Ăž Š ĂŹC5Â&#x;

p−1 2 X

X f (a)

χ(a)e

|L(1, χ)|2m ,

p

χ6=χ0 a=1

Ă‘ ˜‡Â?˜„ (Ă˜. ڌU Â?\Â˜Ă™/ Ăš ˜‡2Ă‚n چDirichlet L-Ÿêƒm ,ÂŤĂŠX. Â?O(/`, òy²eÂĄ ½n: ½ n 5.3 p > 3 ´ÂƒĂŞÂ…χ´ p DirichletA . 2 f (x) =

Pk

i=0 ai x

i

´Â˜Â‡ ĂŞ

XĂŞ kgþ‘ª…ávp ∤ (a0 , a1 , . . . , ak ). KĂŠ?Âż ĂŞm Ăšk, kXeĂŹCúª

p−1 2 X

X f (a)

χ(a)e

|L(1, χ)|2m

p χ6=χ0 a=1 ! m−1 Y 1 − C 2m−2 2− k1 +ÇŤ = p2 Îś 2m−1(2) 1− + O p , p20 p 0

Ù¼

Q

p0

n LÊ¤kĂ˜Ă“up ÂƒĂŞÂŚĂˆ, Cm =

m! , n!(m−n)!

Â…ÂŒO~ĂŞ=Â?6u ĂŞk

Ăš?˜‰½ ¢êǍ. e -f (a) = a, kDirichlet L-Âź ĂŞ †τ (χ)\ Ăž Š ĂŹ C Ăş ÂŞ, Ăš Ă™ ¢ Ă’ ´ Š z[128] ˜‡AĂ?Âœš. r§ ¤XeĂ­Ă˜: Ă­ Ă˜ 5.1 p > 3 ´Â˜Â‡ÂƒĂŞÂ…χ ´ p Dirichlet A , KĂŠ?Âż ĂŞm, kXe ĂŹCúª X

χ6=χ0 2

=p Îś

|τ (χ)|2 |L(1, χ)|2m

2m−1

(2)

Y p0

m−1 1 − C2m−2 1− p20

!

+ O(p1+ÇŤ ).

˜„/, ĂŠuq > 3 ´ ĂŞ, ^8c¤Ă?Âş Â?{Ăƒ{ , ´Âƒ&

q

2 X

X f (a)

χ(a)e

|L(1, χ)|2m ,

q χ6=χ0 a=1

ĂŹCúª´ 3 , Ă?dĂšE´Â˜Â‡m˜5ÂŻK.

Â? ¤½n y², I‡eÂĄA‡Ún. Ă„k‰Ñ‘kþ‘ª n Ăš Â˜Â‡Ă° ÂŞ.

94

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

Ăš n 5.5 f (x) ´ ĂŞXĂŞ kgþ‘ª, PÂ?f (x) = a0 + a1 x + ¡ ¡ ¡ + ak xk , ¿…χ ´ p(p > 3 ´ÂƒĂŞ) DirichletA . Kk

p−1 2 p−1 p−1

X X X f (a)

g(b, a)

χ(a)e χ(a) e ,

=p−1+

p b a=1

a=2

Ù¼g(b, a) = f (ab) − f (b) =

Pk

i=0

b=1

ai (ai − 1)bi .

y²: 5Âż 1 6 b 6 p − 1, Kk(b, p) = 1. ŠâA 5, ÂŒÂą

p−1

2 X p−1

X f (a) f (a) − f (b)

χ(a)e χ(a)χ(b)e ¯

=

p p a=1

a, b=1

=

p−1 X p−1 X a=1 b=1

f (ab) − f (b) χ(ab)χ(b)e ÂŻ . p

g(b, a) = f (ab) − f (b) = Ă?d

k X i=0

ai (ai − 1)bi ,

p−1 2 p−1 p−1

X f (a)

X X g(b, a)

χ(a)e χ(a)e

=

p p a=1 b=1 a=1 X p−1 p−1 p−1 X X g(b, 1) g(b, a) = χ(1)e + χ(a)e p p a=2 b=1 b=1 p−1 p−1 X X g(b, a) =p − 1 + χ(a) e . b a=2 b=1

ÚÒy² Ăšn5.5. Ăš n 5.6 f (x) E , á v Ăš n5.5 ^ ‡, 2 -g(x) = g(x, a) = f (ax) − f (x) = Pk i i i=0 ai (a − 1)x , KkXe OÂŞ 

p−1 1



X ≪ p1− k , p ∤ (b0 , b1 , . . . , bk ) g(b, a)

e

 = p − 1, p|(b0 , b1 , . . . , bk ) p b=1 Ù¼bi = ai (ai − 1), i = 0, 1, . . . , k Â…k ´þԻf (x) gĂŞ.

y²: w, p |(b0 , b1 , . . . , bk )ž(Ă˜¤å. p ∤ (b0 , b1 , . . . , bk )ž, Šâg(x, a) ½

Ă‚, k(ĂŤ Šz[93, 103])

p−1

X 1 g(b, a)

e

≪ p1− k .

p b=1

ÚÒy² Ăšn5.6.

95

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ăš n 5.7 p ´ÂƒĂŞ, p > 3 Â…dm (n) PÂ?rnL¤m‡Ă?f L{‡ê, =dm (n) ´m gĂ?fŸê. KĂŠ?ÂżECĂžs, Re(s) > 1, k ∞ X d2m (n) 1 2m−1 Y 2m−1 A(m, p0 , s), =Îś (s) 1 − s ns p n=1 p 0

(n, p)=1

Q Ù¼Μ(s) ´Riemann zeta-Ÿê¿Â… p0 LÊ¤kĂ˜Ă“up ÂƒĂŞÂŚĂˆ, A(m, p, s) = 2 P2m−2 1 Pi i−j m! j j n i=0 j=0 (−1) C2m−1 Cm+i−j−1 , Cm = n!(m−n)! . pis

y²: ĂŤ Šz[129]ÂĽ Ăšn2.3, Â…-q = p ÂƒĂŞ.

Ăš n 5.8 p ´ ƒ ĂŞp > 3 …χ ´ p DirichletA . PA(y, χ) = A(y, χ, m) = P N6n6y χ(n)dm (n), KkXe O X 4 |A(y, χ)|2 ≪ y 2− 2m +ÇŤ p2 , χ6=χ0

Ù¼Ǎ ´?Âż ½ ĂŞ. y²: ĂŤ Šz[127]ÂĽ Ăšn4, Âż qÂ?ÂƒĂŞ. Ăš n 5.9 p > 3 ´ÂƒĂŞÂ…χ ´ p DirichletA . KĂŠ?Âż1 < a < p Ăš?Âż ĂŞm, kXeĂŹCúª X

χ(a)|L(1, χ)|2m =

χ6=χ0

p dm (a) 2m−1 Y Îś (2) A(m, p0 , s) + O(pÇŤ ), a p 0

Q

Ă™ ÂĽÇŤ ´ ? Âż ½ ĂŞ, L ÂŤ ĂŠ ¤ k Ă˜ Ă“ up ƒ ĂŞ ÂŚ Ăˆ, A(m, p, s) = p0 2 P2m−2 1 Pi i−j m! j j n (−1) C C 2m−1 m+i−j−1 , Cm = n!(m−n)! . i=0 j=0 pis y²: Â? Â?Bü„, X X A(χ, y) = χ(n)dm (n), B(χ, y) = χ(n)dm (n). p 6n6y a

p6n6y

KĂŠRe(s) > 1, Dirichlet ŸêL(s, χ) ýÊĂ‚Ăą, Ă?d|^AbelĂ° ÂŞk m

L (s, χ) =

∞ X χ(n)dm (n) n=1

ns

p

=

a X χ(n)dm (n)

n=1 p

=

ns

X χ(n)dm (n) n=1

ns

96

+s

Z

+s

Z

+∞

A(χ, y) dy y s+1

+∞

B(χ, y) dy. y s+1

p a

p

1ÊÙ Ù§êدK

w,þªés = 1 ¤á(d χ 6= χ0 ). Ïd âDirichlet L-¼ê ½Â, é?¿ êa 6= 1 k X χ(a)|L(1, χ)|2m χ6=χ0

2 ∞

X χ(n)dm (n)

= χ(a)

n n=1 χ6=χ0 ! ∞ ! ∞ X X X χ(n)dm (n) χ(l)d ¯ m (l) = χ(a) n l n=1 χ6=χ0 l=1   ! Z +∞ p X X χ(n)dm (n) Z +∞ A(χ, y) X χ(l)d ¯ (l) B( χ, ¯ y) m = χ(a)  + dy  + dy 2 2 p n y l y p p a χ6=χ0 l=1 16n6  p a  ! p a X X χ(n)dm (n)  X χ(l)d ¯ m (l)  = χ(a) n l n=1 l=1 χ6=χ0  p  Z +∞ a X X χ(n)dm (n)  B(χ, ¯ y)  + χ(a) dy n y2 p n=1 χ6=χ0 ! Z ! p +∞ X X χ(l)d ¯ A(χ, y) m (l) + χ(a) dy p l y2 a χ6=χ0 l=1 ! Z Z +∞ +∞ X A(χ, y) B(χ, ¯ y) χ(a) + dy dy 2 2 p y y p a X

χ6=χ0

≡ M1 + M2 + M3 + M4 .

y35©O Oþª

ª¥ z .

(i)|^ p A 'X, (p, l) = 1 , kð ª   φ(p), en ≡ l mod p ; X χ(n)χ(l) ¯ =  0, Ù¦. χ mod p

K âÚn5.7, éN´  p  ! p a X X χ(n)dm (n) X χ(l)d ¯ (l) m  M1 = χ(a)  n l n=1 χ6=χ0

=

X

l=1

p a

p XX χ(an)χ(l)d ¯ m (n)dm (l)

χ mod p n=1 l=1

nl

97

p

−

p a X X dm (n)dm (l) n=1 l=1

nl

Smarandache ¼ê9Ù '¯KïÄ p

p a X X ′ ′ dm (n)dm (l) = φ(p) + O(pǫ ) nl n=1 l=1 an≡l( mod p) p a 2 X ′ dm (n)dm (a) = φ(p) + O(pǫ ) 2 an n=1 p

a 2 dm (a)φ(p) X ′ dm (n) = + O(pǫ ) 2 a n n=1   ∞ ∞ 2 2 dm (a)φ(p) X ′ dm (n) X ′ dm (n)  = − + O(pǫ ) 2 2 a n n n=1 n= p a  ∞ ∞ 2 2 X d (n) dm (a)φ(p) X ′ dm (n) ′ m  dm (a)φ(p) = + O + pǫ  2 2 a n a n n=1 n= ap p dm (a) 2m−1 1 2m−1 Y ζ = (s) 1 − s A(m, p0 , s) + O(pǫ ). a p

= Ù¥ÎÒ

P′

n

p dm (a) 2m−1 ζ (2) a

p0 6=p

Y

A(m, p0 , s) + O(pǫ ),

p0

Q

L«é¤k pp n¦Ú,

p0

L«é¤kØÓup ê¦È, 3þ

ª¥^ mgØê¼ê {ü O, =dm (n) ≪ nǫ .

(ii) âÚn5.7ÚA 5 , k  p  Z +∞ a X X χ(n)dm (n)  B(χ, ¯ y)  M2 = χ(a) dy 2 n y p n=1 χ6=χ0  p  ! Z p3(2m−2 ) P a X X χ(n)d ¯ χ(n)d (n) m (n) p6n6y m  = χ(a)  dy n y2 p n=1 χ6=χ0  p  Z +∞ P a X X χ(n)d ¯ χ(n)d (n) m (n) m p6n6y  + χ(a)  dy n y2 p3(2m−2 ) n=1 χ6=χ0

p

X

Z p3(2m−2 ) y a X

1

dm (n)dm (l) X

dy ≪ χ(an) χ(l) ¯

y 2

n=1 n p

l=p χ6=χ0 Z +∞ 1 X ǫ +p |B(χ, ¯ y)|dy. 2 p3(2m−2 ) y χ6=χ0

|^CauchyØ ªÚÚn5.8 N´ X

χ6=χ0

1 2

|B(χ, ¯ y)| 6 φ (p) 

1 2

X

χ6=χ0

98

|B(χ, ¯ y)|2

1ÊÙ Ù§êدK

6p

1 2

Ïd, k

M2 ≪

Z

p3(2 p

+p ≪

Z

m−2 )

3 +ǫ 2

Z

p

p ǫ

≪p ,

2

3

2

6 p 2 y 1− 2m +ǫ .

(5-14)

2

3(2m−2 )

p

p

1

p

y a

X X φ(p)

′ ′ dm (n)dm (l)

dy +

y 2 n=1 n l=p

an≡l(p)

+∞

3(2m−2 )

y

2− 24m +ǫ 2

y −1− 2m +ǫ dy p

a ǫ φ(p) X ′ ǫ1 y p y dy + pǫ y 2 n=1 n p a

ùp^ Odm (n) ≪ nǫ . (iii)aq/, ± M3 =

X

χ(a)

χ6=χ0

(iv)éuM4 , k Z X χ(a) M4 =

p X χ(l)d ¯ m (l) l=1

l

! Z

! Z

+∞ p a

A(χ, y) dy y2

!

= O(pǫ ).

B(χ, ¯ z) dz p z2 p a χ6=χ0 ! ! Z p2m−1 Z ∞ ! Z p2m−1 Z ∞ ! X A(χ, y) B(χ, ¯ z) = χ(a) + dy + dz p y2 z2 p2m−1 p p2m−1 a χ6=χ0 ! Z 2m−1 ! Z p2m−1 p X A(χ, y) B(χ, ¯ z) χ(a) = dy dz + p y2 z2 p a χ6=χ0 ! Z Z p2m−1 ∞ X A(χ, y) B(χ, ¯ z) + χ(a) dy dz + p y2 z2 p2m−1 a χ6=χ0 ! Z ∞ Z p2m−1 X A(χ, y) B(χ, ¯ z) + dy dz + χ(a) y2 z2 2m−1 p p χ6=χ0 Z ∞ Z ∞ X A(χ, y) B(χ, ¯ z) + χ(a) dy dz y2 z2 p2m−1 p2m−1 +∞

A(χ, y) dy y2

+∞

χ6=χ0

≡ N1 + N2 + N3 + N4 .

éuN1 , dA(χ, y)ÚB(χ, ¯ z) ½Â±9A Ú 5, k ! Z 2m−1 ! Z p2m−1 p X A(χ, y) B(χ, ¯ z) N1 = χ(a) dy dz 2 2 p y z p a χ6=χ0

99

Smarandache ¼ê9Ù '¯KïÄ

=

Z

p2

m−1

p a

m−1

p

Z

6p

p2

Z

X 1 y2z2 p a

2m−1

2m−1

Z

p p a

p

p

X

6n6y p6l6z

dm (n)dm (l) 

X

χ6=χ0

1 X′ X′ dm (n)dm (l)dydz y 2z2 p a

χ(an)χ(l) ¯  dydz

6n6y p6l6z

an≡l( mod p)

≪ ≪

2m−1

Z Z

p p a

Z

2m−1

p

p

2m−1

p p a

a

2m−1

Z

X 1 dm (n)dydz y 2z 1−ǫ p

p

6n6y

y −1+ǫ z −1+ǫ dydz

p

ǫ

≪p .

éuN2 , âA Ú 5 , ª(5-14)±9 Oªdm (n) ≪ nǫ , ± ! Z Z p2m−1 ∞ X B(χ, ¯ z) A(χ, y) N2 = χ(a) dy dz 2 2 p y z 2m−1 p a χ6=χ0 ! Z 3·2m−2 ! m−1 Z p2 p X B(χ, ¯ z) A(χ, y) χ(a) = dy dz + 2 2 p y z 2m−1 p a χ6=χ0 ! Z Z p2m−1 ∞ X B(χ, ¯ z) A(χ, y) + χ(a) dy dz p y2 z2 p3·2m−2 a χ6=χ0

=

Z

p2

m−1

p3·2

m−2

p2m−1

p a

Z

Z

p2

m−1

Z

X 1 y2z2 p

X

6n6y p6l6z a

∞

dm (n)dm (l)

X

χ(an)χ(l)dydz ¯ +

χ6=χ0

1 X X χ(an)dm (n)B(χ, ¯ z)dydz 2 2 p 3·2m−2 y z p p a χ6=χ0 a 6n6y

Z 2m−1 Z 3·2m−2

p p 1 X′ X′

dm (n)dm (l)dydz + 6 p 2 2

p

y z p p2m−1

a 6n6y p6l6z

a

an≡l( mod p)

Z p2m−1 Z ∞

X X

1 ′

+

d (n) |B( χ, ¯ z)| dydz m

2 2 p 3·2m−2 y z p p

a

χ6=χ0 6n6y a

Z 2m−1 Z 3·2m−2

p

p

≪

y −1+ǫ z −1+ǫ dydz + m−1

p

2 p a

Z 2m−1 Z

p

∞ 3 2

+

y −1+ǫp 2 z −1− 2m +ǫ dydz

m−2

p

p3·2 a m−2 − 2m +ǫ 3 2 ǫ +ǫ 2 ≪p + p p3·2 +

100

1ÊÙ Ù§êدK

≪ pǫ . |^ N2 aq O {, ± N3 ≪ pǫ . éuN4 , dCauchyØ ªÚÚn5.8, ± Z ∞ Z ∞ X A(χ, y) B(χ, ¯ z) χ(a) N4 = dy dz y2 z2 2m−1 2m−1 p p χ6=χ0 Z ∞ Z ∞ 1 X χ(a)A(χ, y)B(χ, ¯ z)dydz = 2 2 p2m−1 p2m−1 y z χ6=χ

0

Z ∞ Z ∞ X

1

dydz 6 χ(a)A(χ, y)B( χ, ¯ z)

2 2 p2m−1 p2m−1 y z χ6=χ

0 Z ∞ Z ∞ 1 X |A(χ, y)| · |B(χ, ¯ z)| dydz 6 2 2 p2m−1 p2m−1 y z χ6=χ 0  1  1 2 2 Z ∞ Z ∞ X 1 X 2 2 6 |A(χ, y)| |B(χ, ¯ z)|  dydz · 2 2 p2m−1 p2m−1 y z χ6=χ0 χ6=χ0 Z ∞ Z ∞ 1 1 1 2− 24m +ǫ 2 2 2− 24m +ǫ 2 2 y p p ≪ · z dydz 2 2 p2m−1 p2m−1 y z Z ∞ Z ∞ 2 2 −1− 22m +ǫ dy · 6p y z −1− 2m +ǫ dz p2m−1

m−1 −

≪ p2 p2 = pǫ .

p2m−1

2 +ǫ 2m

m−1 − 2m +ǫ 2 · p2

ÏddN1 , N2 , N3 ÚN4 O

M4 = O (pǫ ) . nÜ(i), (ii), (iii)Ú(iv) ©Ûá= X p dm (a) 2m−1 Y χ(a)|L(1, χ)|2m = ζ (2) A(m, p0 , s) + O(pǫ ), a p χ6=χ0

Ù¥

Q

p0

0

L«é¤kØÓup ê¦È A(m, p, s) =

2m−2 X i=0

n Cm =

m! n!(m−n)! .

i 2 1 X i−j j j (−1) C C , 2m−1 m+i−j−1 pis j=0

ùÒ ¤ Ún5.9 y².

101

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Ăš n 5.10 p > 3 ´ÂƒĂŞÂ…χ ´ p DirichletA . KĂŠ?Âż ĂŞm k ! m−1 X Y 1 − C 2m−2 |L(1, χ)|2m = p Îś 2m−1(2) 1− + O(pÇŤ ), 2 p 0 p χ6=χ0

0

Ù¼Ǎ ´?Âż ½ ĂŞ,

Q

p0

LÊ¤kĂ˜Ă“up ÂƒĂŞÂŚĂˆ.

y²: ĂŤ Šz[127]ÂĽ Ăšn6, Âż-q ÂƒĂŞ. !ò ¤½n5.3 y². Ă„kdĂšn5.5 k

p−1 2 X

X f (a)

χ(a)e

|L(1, χ)|2m

p χ6=χ0 a=1

=

X

χ6=χ0

! p−1 X g(b, a) p−1+ χ(a) e |L(1, χ)|2m p a=2

= (p − 1) = (p − 1)

p−1 X

X

χ6=χ0

X

χ6=χ0

b=1

|L(1, χ)|

2m

|L(1, χ)|

2m

p−1 X p−1 X g(b, a) X + e χ(a)|L(1, χ)|2m p a=2 χ6=χ0

b=1

p−1 p−1 X X g(b, a) X ′ χ(a)|L(1, χ)|2m + e p a=2 b=1

+

p−1 p−1 X X ′′

χ6=χ0

e

a=2 b=1

Ù¼g(b, a) =

Pk

i

g(b, a) X

p

χ(a)|L(1, χ)|2m ,

χ6=χ0

i

i

− 1)b , bi = ai (a − 1) …

i=0 ai (a

p−1 p−1 X X ′

a=2 b=1

†

p−1 p−1 X X ′′

a=2 b=1

ŠOLÊáv

^‡p ∤ (b0 , b1 , . . . , bk ) Ăšp|(b0 , b1 , . . . , bk ) ÂŚĂš. e¥òŠO OÚ߇ڪ. (1) p ∤ (b0 , b1 , . . . , bk ), ŠâĂšn5.6ÚÚn5.9 k

p−1 p−1

X X X

g(b, a) ′ 2m

e χ(a)|L(1, χ)|

p

a=2 b=1

χ6=χ0

X p−1 p−1 X X

g(b, a)

′

2m

6 e χ(a)|L(1, χ)|

p

a=2 b=1 χ6=χ0

p−1 X X

1 ≪ p1− k +ÇŤ

χ(a)|L(1, χ)|2m

χ6=χ0

a=2 2− k1 +ÇŤ

≪p

p−1 X dm (a) a=2

1

a

≪ p2− k +ÇŤ ,

102

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

Þª^ dm (a) ≪ aÇŤ .

(2) p | (b0 , b1 , . . . , bk )ž, =p | b0 , p | b1 , . . . , p | bk ; dup ∤ (a0 , a1 , . . . , ak ), K–

3˜‡al ÂŚ p ∤ al , Ă?dĂŠuڇl ˜½kp | (al − 1), =Ă“{Â?§ al ≥ 1(modp)

¤å. 38Ăœ{2, 3, ¡ ¡ ¡ , p − 1}ÂĽ, ávp | (al − 1) a ‡ê–þkl − 1 ‡. duÂŽ 1

1

² l − 1 < l 6 k, al > al − 1 > p, u´ka > p l > p k . Ă?ddĂšn5.6ÚÚn5.9 k

p−1 p−1 X

X ′′ X

g(b, a) 2m

e χ(a)|L(1, χ)|

p

a=2 b=1

χ6=χ0

X p−1 p−1 X X

g(b, a)

′′

2m

6 e χ(a)|L(1, χ)|

p

a=2 b=1 χ6=χ0 p−1 X

p dm (a) a a=2 dm (a) 2 max Ă— ♯{a : a ∈ {2, 3, . . . , p − 1}, al ≥ 1(mod p)} ≪p 1/k a p 6a<p

≪

′′

(p − 1)

1

≪ kp2− k +ÇŤ , 1

≪ p2− k +ÇŤ , Ù¼dm (a)E,|^ (1)ÂĽ O Â?{. Ă?d(Ăœ(1), (2)ÚÚn5.10 ĂĄ=ÂŒÂą

p−1

2 X

X f (a)

χ(a)e( ) |L(1, χ)|2m

p

χ6=χ0 a=1 X |L(1, χ)|2m | = |(p − 1) χ6=χ

0

 

p−1 p−1

X

X ′ X

g(b, a) 2m +O 

e χ(a)|L(1, χ)|

 p

a=2 b=1

χ6=χ0

 

p−1 p−1

X X X

g(b, a) ′′ 2m +O 

e χ(a)|L(1, χ)|

 p

a=2 b=1

χ6=χ0 ! m−1 Y 1 − C2m−2 1 = p2 Îś 2m−1(2) 1− + O(p2− k +ÇŤ ), 2 p0 p 0

Ù¼

Q

p0

n LÊ¤kĂ˜Ă“up ÂƒĂŞÂŚĂˆ, Cm =

103

m! n!(m−n)! ,

ÂŒO ~ĂŞÂ?6uk ÚǍ. Ă?d

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

ÏCúª

p−1

2 X

X f (a)

χ(a)e

|L(1, χ)|2m

p χ6=χ0 a=1 ! m−1 Y 1 − C 1 2m−2 = p2 Îś 2m−1(2) 1− + O(p2− k +ÇŤ ). 2 p0 p 0

Ăš Ă’ ¤ ½n5.3 y².

5.3 2Ă‚Dirichlet L-Ÿê q > 3 ´ ĂŞ, …χ´ q DirichletA , ĂŠ?¿¢êa > 0, Ă„Xe½Ă‚ 2 Ă‚Dirichlet L-Ÿê L(s, χ, a) =

∞ X

χ(n) , s (n + a) n=1

Ù¼s = Ďƒ +it ávĎƒ > 0 Ăšt Ă‘´¢ê. |^)Ă›òÿÂ?ÂŒÂąr§*Ă? ‡E²¥(Ă˜ s = 1, χ Â?ĂŒA Âœš). 'u2Ă‚Dirichlet L-Ÿê, Bruce C. Berndt[130−132] ĂŻĂ„

Nþáv, ›^‡ Ă° ÂŞ. Ù¼ Ă?Âś ˜‡´ χ´Â˜Â‡ q A ž, Dirichlet L-ŸêL(s, χ) ávŸêÂ?§ − (s+b) 2 Ď€ 1 Ď„ (χ) (s + b) L(s, χ) = b √ R(1 − s, χ), ÂŻ R(s, χ) = Γ q 2 i q

Ù¼ b= ĂŠĎƒ >

1 2

  0, χ(−1) = 1,

 1, χ(−1) = −1.

− m Â…m´ ĂŞ, Berndt[132]

  m−1 −s j X a  (−1) Γ(s + j)L(−j, χ) L(s, χ, a) = + G(s) , Γ(s) j=0 j! aj

Ù¼G(s) ´)Ă›Ÿê. n Â?š êž, N´OÂŽĂ‘L(n, χ, a), AO/, L(0, χ, a) = L(0, χ). Ă“ ž, ' uDirichlet L-Âź ĂŞ Ăž Š 5 Â&#x; Â? k N Ăľ Æ Ăś Ă‘ Š ĂŻ Ă„(ĂŤ Š z[133-136]). qÂ?ÂƒĂŞpž, I. Sh. Slavutskiˇi[133] 31986c‰Ñ

X Ď€2 |L(1, χ)|2 = p − log2 p + θ log p, 6 χ6=χ0

Ù¼, p 6 35ž, |θ| < 10. 104

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

ĂœŠ+[136] KĂŻĂ„ ĂŠÂ˜Â„ q Dirichlet L-Ÿê ފ5Â&#x;,

 2 2 2 X Y X log p Ď€ 1 φ (q)  + O(log log q), |L(1, χ)|2 = φ(q) 1 − 2 − 2 log q + 6 p q p−1

χ6=χ0

Ù¼φ(q)´EulerŸê,

p|q

p|q

P

p|q LÂŤĂŠq Ă˜Ă“ÂƒĂ?fÂŚĂš, [137]

D.R. Heath-Brown

Q

p|q LÂŤĂŠq Ă˜Ă“ÂƒĂ?fÂŚĂˆ. 1 Ăž ފ5Â&#x; 2

ĂŻĂ„ Dirichlet L-Ÿê3s =

X 1

2 φ(q) X q

L , χ

= Âľ T (k),

2 q k

χ mod q

k|q

Ù¼T (k) kĂŹCúª 2 2N−1 X 1 n k 1 k2 + cn k − 2 + O(k −n ), T (k) = log + Îł + 2Îś 8Ď€ 2 n=0 ávN > 1,cn ´ÂŒOÂŽ ~ĂŞÂ…γ´Euler~ĂŞ. Ă˜dƒ , R. Balasubramanian[138] Dirichlet L-Ÿê3Ďƒ = 12 ‚Þ ĂŹCúª

2 2 X 1 √ √

L

= φ (q) log(qt)+O(q(log log q)2 )+O(te10 log q )+O(q 12 t 23 e10 log q ), + it, χ

2 q

χ mod q

Ù¼t > 3 Â…ʤk q Ă‘¤å.

3 !ÂĽ, XJ s = 1, a > 1, Êڇ2Ă‚Dirichlet L-Ÿê ފš~a, , = Ž‡ X

χ6=χ0

| L(1, χ, a)|2

ĂŹCúª, Ù¼χ ´ q DirichletA …χ0 LÂŤĂŒA . ,˜Â?ÂĄ, 2Ă‚iĂšb Â?Ă‘Dirichlet L-Ÿê ¤kšw,":Ă‘ uÂ†Â‚Ďƒ = 12 Ăž. ĂšÂ˜ÂŻKĂĄĂš {¤Þ¯ Ăľ#ÑÆÜ ĂŻĂ„, ¿…Œ‚¼k˜ Ž² 'uDirichlet L-Ÿê ":ŠĂ™ ­Â‡5Æ, ´EĂŽvkˆ <‚¤ýĂ? @ {. Ă?dĂšÂ˜ÂŻK´ÂŠ UYĂŻĂ„ ˜‡­Â‡Â‘K. !K^Â˜ÂŤuĂ‘ Ăş1ĂŻĂ„2Ă‚Dirichlet L-Ÿê3Ďƒ = 12 ‚Þ ފ 5Â&#x;, l ‰Ñ

2 X 1

L

+ it, χ, a

2

χ mod q

,Â˜Â‡ĂŹCúª, Ù¼χ ´ q DirichletA , 0 6 a 6 1. 'u2Ă‚Dirichlet L-Ÿê ފ5Â&#x; ĂŻĂ„, 8c¤ $ . – „vkĂŠ ? Ă›Âƒ' ĂŤ Šz. ÂŚ+XdÆ܂Â?,ĂŠa, , Ă?Â?ږ ÂŒ¹Ê 2Ă‚Dirichlet L-Ÿê†Dirichlet L-Ÿê ƒm ,ÂŤĂŠX. =y² Xe ߇½n: ½ n 5.4 q > 3 ´ ê…χ ´ q DirichletA . Hurwitz ̟ê½Ă‚Xe, ĂŠu? 105

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Âż EĂŞs, k Îśs, a =

∞ X

1 (n + a)s n=0

(a > 0, s = Ďƒ + it, Ďƒ > 1).

(5-15)

KÊ?¿ ¢êa > 1, kXeÏCúª: a

X

χ6=χ0

[d] X Âľ(d) a 4φ(q) X Âľ(d) X 1 φ(q) log q 2 |L(1, χ, a)| = φ(q) Îś 2, − +O , √ d2 d a d k q d|q

k=1

d|q

Ù¼φ(q) ´EulerŸê, Âľ(d) ´M¨obiusŸê, ÂŒO~ĂŞ=Â?6ua. w,, d > až, ½n5.4ÂŒ{zÂ? X X Âľ(d) a φ(q) log q 2 |L(1, χ, a)| = φ(q) Îś 2, +O ; √ d2 d q χ6=χ0

d|q

d = a ž, ½n5.4=CÂ? X Ď€ 2φ(q) Y 1 4φ2 (q) φ(q) log q 2 |L(1, χ, a)| = 1− 2 − +O . √ 6 p aq q χ6=χ0

p|q

^aqÂ?{, Â?ÂŒÂą 'u2Ă‚Dirichlet L-Ÿê 2k (k > 2)gފ X |L(1, χ, a)|2k . χ6=χ0

½ n 5.5 q > 3 ´ ĂŞÂ…t > 3 ´¢ê, χ ´ q DirichletA . KĂŠ?Âż ¢ ĂŞ0 6 a 6 1, keÂĄ ĂŹCúª  

2 2 X 1 X X Âľ(d)

log p 

L

= φ (q) log qt + 2Îł + + it, χ, a βa/d − φ(q)

2 q 2Ď€ p−1 d

χ mod q

Ù¼φ(q) ´EulerŸê, βx = ÂŤq Ă˜Ă“ÂƒĂ?f ‡ê.

p|q

d|q

5 1 1 5 +O qt− 12 + t 6 log3 t2ω(q) + q 2 t 12 log t2ω(q) , P∞

x n=1 n(n+x)

´Â?6ux ˜‡ŒOÂŽ ~ĂŞ, …ω(q) L

w,, a = 0ž, ĂĄ=ÂŒÂą Dirichlet LŸê3Ďƒ = 12 ‚ÞÞŠ ĂŹCúª. Ă?d ĂšÂ˜(J´cÂĄ(J Â˜Â‡Ă­2. ĂŠu2Ă‚Dirichlet L-Ÿê 2k (k > 2) gފ

2k X 1

L + it, χ, a

,

2 χ mod q

É8cÂ?{  Â›Â„Ăƒ{ § ĂŹCúª. $–Êk = 2, Â?Ăƒ{Âź ˜‡Ă? ĂŹC úª. Ă?dĂšI‡?Â˜Ăšg Â?\k Â?{.

106

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

5.3.1 'u½n5.4 Â? y²½n5.4, I‡eÂĄ A‡Ún. Ă„k‰ÑDirichlet L-Ÿê†Ù2Ă‚/ÂŞ Â˜Â‡Ă° ÂŞ. Ăš n 5.11 q > 3 ´ ĂŞ, …χ ´ q DirichletA . L(s, χ) Â?ĂŠAA Â?χ Dirichlet L-Ÿê, L(s, χ, a) Â?2Ă‚Dirichlet LŸê. KĂŠ?Âż ¢êa > 0, k L(1, χ, a) = L(1, χ) − a

∞ X

χ(n) . n(n + a) n=1

y²: ŠâDirichlet L-Ÿê9Ă™2Ă‚/ÂŞ ½Ă‚, k ∞ ∞ X χ(n) X χ(n) − n + a n=1 n n=1 ∞ X χ(n) χ(n) − = n+a n n=1

L(1, χ, a) − L(1, χ) =

= −a

∞ X

χ(n) , n(n + a) n=1

£‘ L(1, χ, a) = L(1, χ) − a ÚÒy² Ăšn5.11.

∞ X

χ(n) . n(n + a) n=1

Ăš n 5.12 q ´ ĂŞÂ…q > 3 …χ ´ q DirichletA . PA(y, χ) = KkXe O X

χ6=χ0

P

N6n6y

χ(n)d(n).

|A(y, χ)|2 ≪ yφ2(q),

y²: ĂŤ Šz[127]ÂĽ Ăšn4, Â…-k = 2. Ăš n 5.13 q > 3 ´ ê…χ ´ q DirichletA . Kk X Y 1 2 |L(1, χ)| = φ(q)Îś(2) 1 − 2 + O(q ÇŤ), p χ6=χ0

Ù¼φ(q) ´EulerŸê, Â…

p|q

Q

p|q

LÂŤĂŠq ¤kĂ˜Ă“ÂƒĂ?fÂŚĂˆ, ÇŤ ´?¿‰½ ĂŞ.

y²: ĂŤ Šz[127]ÂĽ Ăšn6Âż-k = 2. Ăš n 5.14 q > 3 ´ ê…χ ´ q DirichletA . KĂŠ?Âż ¢êa > 1, k ∞ XX

χ6=χ0

χ(n) ¯ L(1, χ) n(n + a) n=1 107

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„ [ ad ] Y 1 φ(q) log q φ(q) 1 φ(q) X Âľ(d) X = 1− 2 + 2 . Îś(2) +O √ a p a d k q p|q

k=1

d|q

y²: Ă„kA^AbelĂ° ÂŞk L(1, χ) =

q X χ(n) n=1

n

+

Z

+∞

q

A(χ, y) dy, y2

∞ X

Z +∞ N X χ(n) ÂŻ χ(n) ÂŻ (2y + a)A(χ, ÂŻ y) dy, = + 2 n(n + a) n=1 n(n + a) y (y + a)2 N n=1 P Ù¼A(χ, y) = q<n<y χ(n), Â…N > q ´ ĂŞ. ddk ∞ XX

χ6=χ0

=

χ(n) ¯ L(1, χ) n(n + a) n=1 N X

χ(n) ¯ + n(n + a) n=1

X

χ6=χ0

Z

+∞

N

(2y + a)A(χ, ¯ y) dy y 2 (y + a)2

!

q X χ(n) n=1

n

+

Z

+∞

q

A(χ, y) dy y2

!

Z +∞ q N X XX χ(n) ÂŻ χ(m) χ(n) ÂŻ A(χ, y) + = dy + 2 n(n + a) m n(n + a) y q m=1 χ6=χ0 n=1 χ6=χ0 n=1 Z q +∞ X X χ(n) (2y + a)A(χ, ÂŻ y) + dy + n y 2 (y + a)2 N χ6=χ0 n=1 Z +∞ X Z +∞ (2y + a)A(χ, ÂŻ y) A(χ, z) dy dz + y 2 (y + a)2 z2 N q N XX

χ6=χ0

≥ A1 + A2 + A3 + A4 . ò OÞªÂ

˜‡ Ă’ÂĽ z˜‘. Ă„k OA1 . dA 5, CauchyĂ˜

ªÚÚn5.13, A1 =

χ6=χ0

=

q X χ(n) ¯ χ(m) n(n + a) m=1 m n=1

N XX

X

χ mod q

=

q N X X χ(m)χ(n) ¯ + O (log q) mn(n + a) n=1 m=1

q N X X ′

X 1 χ(m)χ(n) ¯ + O (log q) mn(n + a)

′

n=1 m=1

= φ(q)

χ mod q

q N X X ′

′

n=1 m=1 m≥n( mod q)

= φ(q)

q X

1 + O (log q) mn(n + a) N/q

′

m=1

q

X X 1 1 ′ ′ + φ(q) + O (log q) 2 m (m + a) m(kq + m)(kq + m + a) m=1 k=1

108

1ĂŠĂ™ Ă™§êĂ˜ÂŻK q X

X 1 Âľ(d) + 2 (m + a) m m=1 d|(m,q)   N/q q X X 1 ′ ′ +O φ(q) + log q  m(kq + m)(kq + m + a) m=1

= φ(q)

k=1

= φ(q)

X

Âľ(d)

1 + O (log q) + a/d)

d3

m2 (m

1 + + a/d)

d3

m=1

∞ X Âľ(d) X d|q

+O φ(q)

d|q

d3

X Âľ(d) d|q

m=1



∞ X Âľ(d) X

X Âľ(d) d|q

= φ(q)

+ O (log q)

m2 (m



= φ(q)

+ a)

q/d X Âľ(d) X d|q

= φ(q)

1

m2 (m

m=1 d|m

d|q

= φ(q)

q X

d3

1  + O (log q) d3 m2 (m + a/d) m=q/d ! ∞ 1 1 X a/d 1 + + O (log q) − (a/d)2 m=1 m2 m + a/d m ! ∞ X d d2 1 1 + − + O (log q) am2 a2 m + a/d m m=1

[a/d] φ(q) X ¾(d) X 1 φ(q) X ¾(d) Μ(2) + 2 = + O (log q) a d2 a d k k=1

d|q

d|q

[a/d] Y 1 φ(q) X Âľ(d) X 1 φ(q) Îś(2) + O (log q) . = 1− 2 + 2 a p a d k p|q

k=1

d|q

eÂĄUY OA2 , A3 ĂšA4 . ŠâĂšn5.12, 'uA Ăš P´olyaĂ˜ ÂŞÂą9CauchyĂ˜ ÂŞ, |^{Ăź O, k |A2 | =

N XX

χ6=χ0

χ(n) ¯ n(n + a) n=1

Z

N XX

+∞

q

Z +∞ 1 6 n(n + a) q χ6=χ0 n=1 Z ∞ 1 1 2 ≪ q φ(q) log q dy y2 q φ(q) log q ≪ , √ q

109

A(χ, y) dy y2

P

q<n<y χ(n)

y2

dy

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Z q XX χ(n)

|A3 | =

χ6=χ0 n=1 Z ∞

≪ log q

≪ log q ≪ |A4 | = 6

q

∞Z ∞

N

q

2

≪

+∞

χ6=χ0 N Z ∞Z ∞

Z

Z

∞

N

N

(2y + a)A(χ, ¯ y) dy y 2 (y + a)2

2y + a X |A(χ, ÂŻ y)|dy y 2 (y + a)2 χ6=χ0  1  1 2 2 X X 2y + a  2 1  |A(χ, ÂŻ y)|  dy y 2 (y + a)2 χ6=χ0

χ6=χ0

φ (q) log q √ , N N

XZ N

6

3 2

N

n

+∞

(2y + a)A(χ, ¯ y) dy 2 y (y + a)2

Z

+∞ q

A(χ, z) dz z2

X 2y + a |A(χ, ÂŻ y)| ¡ |A(χ, z)|dydz y 2(y + a)2 z 2 χ6=χ0  1  1 2 2 X X 2y + a 2 2  |A(χ, ÂŻ y)|   |A(χ, z)|  dydz y 2(y + a)2 z 2 χ6=χ0

χ6=χ0

φ (q) √ √ , N N q

N = q 2 , u´k [ ad ] Y χ(n) ÂŻ φ(q) 1 φ(q) X Âľ(d) X 1 φ(q) log q L(1, χ) = Îś(2) 1− 2 + 2 +O . √ n(n + a) a p a d k q n=1

∞ XX

χ6=χ0

p|q

k=1

d|q

Ă?dy² Ăšn5.14. Ăš n 5.15 q > 3 ´ ê…χ ´ q DirichletA , KĂŠ?Âż ¢êa > 1, kXeĂŹ Cúª

2 ∞ X

X Y χ(n)

φ(q) 1 φ(q) X Âľ(d) a = Îś(2) 1 − + Îś 2,

2 2 2 2

n(n + a) a p a d d n=1

χ6=χ0

p|q

d|q

[ ad ]

2φ(q) X Âľ(d) X 1 +O − 3 a d k k=1

d|q

φ(q) log q q

.

y²: N > q Â??˜ ĂŞ, |^A 5, P´olyaĂ˜ ÂŞÂą9Ăšn5.12, k

∞

2 X

X χ(n)

n(n + a)

n=1 χ6=χ0 ! Z +∞ N X X χ(n) (2y + a)A(χ, y) = + dy Ă— n(n + a) y 2 (y + a)2 N n=1 χ6=χ0

110

1ÊÙ Ù§êدK

! (2z + a)A(χ, ¯ z) dz z 2 (z + a)2 N ! N X χ(m) ¯ + m(m + a) m=1 χ6=χ0 ! Z N +∞ X X χ(n) (2z + a)A(χ, ¯ z) dz + + n(n + a) z 2 (z + a)2 N χ6=χ0 n=1 ! N X Z +∞ (2y + a)A(χ, y) X χ(m) ¯ + dy + 2 2 y (y + a) m(m + a) N m=1 χ6=χ0 X Z +∞ (2y + a)A(χ, y) Z +∞ (2z + a)A(χ, ¯ z) + dy dz y 2 (y + a)2 z 2 (z + a)2 N N χ6=χ0   N N X X X 1 1  + χ(n)χ(m) ¯ = n(n + a) m(m + a) n=1 m=1 χ6=χ0 P Z +∞ (2y + a) χ6=χ0 |A(χ, y)| dy +O y 2(y + a)2 N ! 3 N X N X 2 (q) 1 φ ′ ′ = φ(q) + O(1) + O 3 mn(m + a)(n + a) N2 n=1 m=1 N X

χ(m) ¯ × + m(m + a) m=1 ! N X X χ(n) = n(n + a) n=1

Z

+∞

m≡n( mod q)

= φ(q)

N X

′

n=1

= φ(q)

N X

′

n=1

N X

1 + 2φ(q) 2 n (n + a)2 

N X N X ′

′

n=1 m=1 m≡n( mod q), m>n N/q

X 1 ′ φ(q) + O n2 (n + a)2

N X

′

k=1 n=1

1 + O(1) mn(m + a)(n + a) 

1  + O(1) n(n + a)(kq + n)(kq + n + a)

X 1 φ(q) log N = φ(q) µ(d) + O + O(1) n2 (n + a)2 q n=1

= φ(q)

N/d X µ(d) X d|q

= φ(q)

d|(n,q)

d4

∞ X µ(d) X d|q

d4

+O φ(q) = φ(q)

1 +O 2 n (n + a/d)2 n=1

X µ(d) d|q

d4

φ(q) log N q

1 + 2 (n + a/d)2 n n=1

∞ X µ(d) X d|q

d4

n=N/d

∞ 3 X

d a3

n=1

1 +O 2 n (n + a/d)2

φ(q) log N q

a/d a/d 2 2 + 2 + − 2 (n + a/d) n n + a/d n

111

!

+O

φ(q) log N q

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Y φ(q) 1 = 2 Îś(2) 1− 2 + a p p|q

[a]

d φ(q) X Âľ(d) a 2φ(q) X Âľ(d) X 1 + 2 Îś 2, − +O a d2 d a3 d k

d|q

k=1

d|q

φ(q) log N q

.

N = q 2 , K

2 ∞ X

X Y χ(n)

φ(q) 1 φ(q) X Âľ(d) a 1− 2 + 2 Îś 2,

= 2 Îś(2) 2

n(n + a) a p a d d n=1 χ6=χ0

p|q

d|q

[ ad ]

2φ(q) X Âľ(d) X 1 +O − 3 a d k k=1

d|q

φ(q) log q q

.

Ăš Ă’y² Ăšn5.15. !ĂŒÂ‡ ¤½n5.4 y². dĂšn5.11, Ăšn5.12, Ăšn5.13, ÂŒ¹å= ½ n5.4. =ĂŠ?Âżq > 3, k X |L(1, χ, a)|2 χ6=χ0

2 ∞ X

X χ(n)

=

L(1, χ) − a

n(n + a)

n=1

χ6=χ0

∞ XX

χ(n) ÂŻ L(1, χ) − n(n + a) n=1 χ6=χ0 χ6=χ0

2 ∞ ∞

X X χ(n) X

X χ(n)

L(1, χ) ÂŻ + a2 −a

n(n + a) n(n + a) χ6=χ0 n=1 χ6=χ0 n=1   Y 1 = φ(q)Îś(2) 1 − 2 + O(q ÇŤ ) p p|q   [ ad ] Y X X φ(q) 1 φ(q) Âľ(d) 1 φ(q) log q  −2a  Îś(2) 1− 2 + 2 +O √ a p a d k q k=1 p|q d|q  Y 1 φ(q) X Âľ(d) a 2  φ(q) +a Îś(2) 1 − + Îś 2, a2 p2 a2 d2 d p|q d|q  [ ad ] X X 2φ(q) Âľ(d) 1 φ(q) log q  − 3 +O a d k q =

X

2

|L(1, χ)| − a

d|q

k=1

a

[d] X Âľ(d) a 4φ(q) X Âľ(d) X 1 φ(q) log q = φ(q) Îś 2, − +O . √ d2 d a d k q d|q

k=1

d|q

112

1ÊÙ Ù§êدK

ùÒ ¤ ½n5.4 y².

5.3.2 'u½n5.5 Äk{ü0 e Ù=ò ^ ­ { . ù {¡ 7:{, § ´van der Corput[139] u20­V20c ^5?n êÈ© êÚ , Ï êØ¥Nõ ­ ¯K )ûI éù êÚ êÈ©k Ð ìCúª½öÐ O, ù { Ï~ < ¡ van der Corput {. !3½n5.5 y²¥, 2 /A^

ù {. ±eA 5 {ü V) ù {: 5 5.1µ µ f (x), f ′ (x), g(x), g ′ (x) ´«m[a, b] þ ¢ üNëY¼ê, ¿ |f ′ (x)| 6 δ < 1, |g(x)| 6 h0 , |g ′(x)| 6 h1 . Kk X

g(n)e(f (n)) =

Z

b

g(x)e(f (x))dx + O

a

a<n6b

h0 + h1 1−δ

,

(5-16)

Ù¥e(f (x)) = exp(2πif (x)), O ~ê´ýé . 5 5.2µ µ 35 5.1 ^ e, g(x)/f ′(x) 3«m[a, b] þ´üN , ÷vf ′ (x)/g(x) > m > 0 ½−f ′ (x)/g(x) > m > 0. Kk

Z b

4 if (x)

g(x)e dx

6 .

m a

(5-17)

5 5.3µ µ 35 5.1Ú5 5.2 ^ e, k

X

≪ 1 + h0 + h1 . g(n)e(f (n))

m 1−δ

a<n6b

(5-18)

(Ü5 5.1Ú5 5.2, á=

, , ò^ êénØ ' £. êéÌ ´?n/X X S= e (f (n)) (B > 1, 1 < h 6 B) B<n6B+h

êÚ O, ± S Ð þ.. F" Xe/ª 'uS þ. S ≪ Aκ B λ , eA ≪ |f ′ (x)| ≪ A (A > 12 ), 0 6 κ 6

1 2

6 λ 6 1, ù | K¢ê(κ, λ)Ò¡ ê

é.

w ,, (κ, λ) = (0, 1) ´ ê é, ¿ N ê é | ¤ 8 Ü ´ à 8, P ∆ = {(tκ1 + (1 − t)κ2 , tλ1 + (1 − t)λ2 ) |(κ1 , λ1 ) (κ2 , λ2 )´ êé, 0 6 t 6 1}. ù

113

Smarandache ¼ê9Ù '¯KïÄ

(JÑ´é² , XJ ¦f (x) r (r > 5) ê÷v: AB 1−r ≪ |f (r) (x)| ≪ AB 1−r

(r = 1, 2, 3, · · · ),

K ± 'uS ² (J. ~X, |^f (x) 2 , kXe5 : 5 5.4µ µ h > 1, f ′′ (x) 3«m[B, B + h] ´ëY ÷v 0 < λ 6 |f ′′ (x)| 6 Cλ, Kk X

1 2

− 12

e(f (n)) = O Chλ + λ

B<n6B+h

,

Ù¥ O ~ê´ýé . l5 5.4 á= ± êé( 12 , 12 ), = S ≪ B AB −1

12

+ AB −1

− 12

1

1

≪ A2B 2.

Ød , ±|^e¡ ü 5 E¤I êé. 5 5.5µ µ e(κ, λ)´ êé, K (µ, ν) =

κ 1 λ , + 2κ + 2 2 2κ + 2

´ êé. 5 5.6µ µ e(κ, λ)´ êéÓ ÷v^ 3 κ + 2λ > , 2 K (µ, ν) =

1 1 λ − ,κ+ 2 2

´ êé. Ïd ±ÏL5 5.5Ú5 5.69c¡¤` êé à55 E¤I ê é. ~X, êé( 12 , 12 ) ±@ |^5 5.6d êé(0, 1) , |^5 5.5é êé( 12 , 12 ) ^ , q # êé( 16 , 23 ). , ± E¦^5 5.5, 5 5.6± 9 êé à5. X êé( 27 , 47 ) Ò´é êé( 16 , 23 ) kA^5 5.5, U 2A^5 5.6 . êénØ3¢ ¥A^ ~2 , Ù¥^ êé= ud, ' u õ êé ?Ø ±ë ©z[139]. Ø þ¡ êénØ, I ^ 'uHurwitz ζ¼ê ìC¼ê § X 1 x1−s x−σ ζ(s, a) = − +O , (n + a)s 1 − s 1 − C −1 06n6x−a

114

(5-19)

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

džxávx >

Ct∗ ∗ 2Ď€ (t

= max(2Ď€, |t|)), Â…C > 1Â?~ĂŞ. a = 1žÂ?ÂŒÂą Riemann

̟ê ĂŹCúª. Ă™ĂŒÂ‡|^XeĂŹCúª.

5 Â&#x; 5.7Âľ Âľ s = Ďƒ + it, Â…¢êx, y > C > 0 áv2Ď€xy = t, KĂŠu0 < Ďƒ1 6 Ďƒ 6 Ďƒ2 < 1, t > 0, k X

Îś(s, a) =

X e(−av) 1 + A(s) + (n + a)s v 1−s

06n6x−a

+O(x

âˆ’Ďƒ

16v6y

log(y + 2) + y

Ďƒâˆ’1

1

(|t| + 2) 2 âˆ’Ďƒ ),

Ă™ÂĽÂŒO~ĂŞÂ?6uĎƒ1 , Ďƒ2 , Â…

1 âˆ’Ďƒ

t 2

t π 1

A(s) =

exp −i t log

− 1+O

2Ď€ 2Ď€e 4 |t|

(5-20)

(t → +∞).

t 6 0, KÂŞ(5-20)CÂ? Îś(s, a) =

X

06n6x−a

X e(av) 1 ′ + A (s) + (n + a)s v 1−s 16v6y

1

+O(xâˆ’Ďƒ log(y + 2) + y Ďƒâˆ’1(|t| + 2) 2 âˆ’Ďƒ ), Ù¼

1 âˆ’Ďƒ

t 2

t π 1

+ A (s) =

exp −i t log

1+O

2π 2πe 4 |t| ′

(t → −∞).

AO/, -s = 12 + it, KĂŠ?Âżt > 0, k 1 Îś + it, a 2 it X X e(−av) Ď€ 1 1 2Ď€ = + ei(t+ 4 ) + O(x− 2 log t). 1 1 +it −it t 2 2 06n6x−a (n + a) 16v6y v

(5-21)

Â? y²½n5.5, „I‡eÂĄ A‡Ún. Ăš n 5.16 q ´ ĂŞ, k X Âľ(d) log d d|q

d

=−

φ(q) X log p . q p−1 p|q

y²: ĂŤ Šz[135]ÂĽĂšn2. Ăš n 5.17 q ´Â˜ ĂŞ, Â…0 6 a 6 1, t > 2 ´¢ê, χ ´ q DirichletA . K kĂ° ÂŞ

2 2 q/d X 1 X X

φ(q) 1 b + a/d

L

Îś

, ¾(d) + it, χ, a

= + it,

2 q 2 q/d

χ mod q

d|q

b=1

Ù¼Μ(s, Îą) ´Hurwitz ̟ê, §ÂŒÂą)Ă›òÿ ²¥(s = 1:Ă˜ ). 115

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

y²: s = Ďƒ + it, Â…Ďƒ > 1, K2Ă‚Dirichlet L-ŸêýÊĂ‚Ăą, Ă?dk q q ∞ ∞ X X X χ(n) χ(b) 1 X b+a L(s, χ, a) = = = s χ(b)Îś s, . (5-22) (n + a)s (nq + b + a)s q q n=1 n=0 b=1

b=1

|^ q A Ăš 5, dÂŞ(5-22)

q

2

X X X 1 b + a

|L(s, χ, a)|2 = 2Ďƒ χ(b)Îś s,

q q χ mod q χ mod q b=1 q X q 1 X b1 + a X ÂŻ b2 + a = 2Ďƒ Îś s, Îś s, χ(b1 )χ(b ÂŻ 2) q q q b1 =1 b2 =1 χ mod q q q φ(q) X X b1 + a ÂŻ b2 + a = 2Ďƒ Îś s, Îś s, q q q b =1 b =1 1 2 (b1 , q)=1 (b2 , q)=1

φ(q) = 2Ďƒ q

b1 ≥b2 ( mod q)

q X

b=1 (b, q)=1

2

Îś s, b + a

q

q φ(q) X

b + a

2 X = 2Ďƒ Âľ(d)

Îś s, q

q b=1

d|(b, q)

q/d X

φ(q) X b + a/d

2

= 2Ďƒ Âľ(d)

Îś s, q/d

, q d|q

b=1

|^Μ(s, a) ÚL(s, χ, a) )Û5, -s =

1 2

+ it, k

2 2 q/d X 1 X X

φ(q) 1 b + a/d

L

Îś

=

. + it, χ, a + it, ¾(d)

2 q 2 q/d

χ mod q

d|q

b=1

Ăš Ă’y² Ăšn5.17.

ÂąeÂ? Â?B, Pa′ = a/d, q ′ = q/d. Ăš n 5.18 q ′ > 2 ´ ĂŞ, Â…0 6 a′ 6 1, t > 3. Kk

!

q′ ′ X

1 1 b+a 1

≪ q ′ 12 t 125 log t + q ′ t− 14 log t,(5-23) Îś − it, − 1 1 ′ ′

′ +it −it

2 q 2 2 ( b+a

b=1 ( b+a

q′ ) q′ ) Ù¼≪ ~ê=�6ua′ .

y²: Šâ5Â&#x;5.7, k 1 b + a′ 1 Îś − it, − 1 −it 2 q′ b+a′ 2 q′

116

1ĂŠĂ™ Ă™§êĂ˜ÂŻK

=

1

X

b+a′ q′ )

(n +

16n6x

ddĂĄ=

q′

X 1

b+a′ 12 +it

b=1 ( q′ ) ′

=

q X

1

b=1 ( 

Ă—

b+a′ q′

1

) 2 +it

X X

16n6x b=1

+O(q ′ x

Ă—

(n +

(

− 21

2Ď€ t

b+a 1 − it, 2 q′

b+a′ 21 −it ) q′

n+(b+a′ )/q ′ (b+a′ )/q ′

q′

+

−it

′

1

X

16n6x

=

Îś

1 −it 2

b+a′ q′

1 2

+

it

b+a′ q′

) (n +

)

1 2

2Ď€ t

+

X e

16v6y

′ − b+a v q′ 1 +it 2

v

1

+ O(x− 2 log t),

!

1 − b+a′ 1 −it

( q′ ) 2

−it

Ď€

ei(−t+ 4 )

2Ď€ t

Ď€

ei(−t+ 4 )

b+a′ X e − q′ v

16v6y

−it

v

1 +it 2



1 + O(x− 2 log t)

′ b+a′ e − ( b+a v v)−it X X q′ q′ q′

Ď€

ei(−t+ 4 )

′

1

2 ( b+a q ′ v)

16v6y b=1

+

log t) 1

≥ M1 + M2 + O(q ′ x− 2 log t),

(5-24)

Ù¼x, y ´Â–½~ĂŞ, Â…2Ď€xy = t, 0 < C < x, y < t. 3M1 ÂĽ, b+a′ q′ b+a′ q′

n+

Ù¼Pf (b) =

t 2Ď€

!it

′

=

nq ′ + b + a′ b + a′

it

= eit log

′

+b+a log nqb+a . , , g(b) = ′

Ă˜5 OM1 , Ă?dk

nq ′ +b+a′ b+a′

1

b+a′ q′

1 2

n+ b+a q′

′

1 2

t −nq ′ f (b) = < 0, 2Ď€ (b + a′ )(nq ′ + b + a′ ) ′

g ′ (b) =

−q ′ (nq ′ + 2b + 2a′ ) . (b + a′ )(nq ′ + b + a′ )

= e2Ď€if (b) ,

, |^5Â&#x;5.3ĂšÂ?ĂŞĂŠn

(5-25)

(5-26)

b + a′ > t ž, |f ′ (b)| 6

1 < 1, 2Ď€

1

1

(5-27) 1

1

|g(b)| 6 q ′ t− 2 (nq ′ )− 2 = q ′ 2 (nt)− 2 ,

(5-28)

|g ′(b)| ≪ qt−1 ,

(5-29)

117

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„ 1

1

tnq ′ (b + a′ ) 2 (nq ′ + b + a′ ) 2 f ′ (b) − = g(b) 2Ď€(b + a′ )(nq ′ + b + a′ ) q′ tn

1

tn 2 = . 1 1 > 1 1 > q′ 2π(b + a′ ) 2 (nq ′ + b + a′ ) 2 2π(q ′ + 1) 2 (nq ′ + q ′ + 1) 2

tn

b + a′ < t ž, ÊuN < b < N + h < 2N, …|f ′ (b)| ≪

t N,

|^Â?ĂŞĂŠ

1 X

1 1 t 6 2

e (f (b)) ≪ N 3 ≪ t6 N 2 .

N

N6b6N+h62N

Ă?d, du5Â&#x;5.3Úª(5-27)-(5-31),

X X

|M1 | 6 g(b)e (f (b))

16n6x 16b6q ′

X

X X

6 g(b)e (f (b)) +

′

(5-30) 1 2 6, 3

O

(5-31)

k

X

g(b)e (f (b))

16n6x b+a >t 16n6x b+a′ 6t

′

X q X X X

1 1

− ′

1 + (nt) 2 q 2 +

g(b)e (f (b))

≪

2

16n6x n t 16n6x j≪log t 2j−1 <b<2j

X

1 1 1 X q′ X X

′2 −2 2

≪ + O q t x + max g(b)e (f (b))

1

′ 16N6t−a

2t n

16n6x 16n6x j≪log t N<b<2N 1

1

1

1

1

1

1

1

1

1

1

≪ q ′ x 2 t−1 + q ′ 2 x 2 t− 2 +

X

X 1 1

max ′

+ e (f (b))

log t max ′ N+a′ 1 ′ 1 N+a

16N6t−a (

2 2 16N6t−a

16n6x N<b<N +h<2N q ′ ) (n + q ′ )

′ X X 1 1 1 1 q

≪ q ′ x 2 t−1 + q ′ 2 x 2 t− 2 + log t max ′ √ √ √

e (f (b))

16N6t−a N n q ′

16n6x N<b<N +h<2N

X √q ′ X

1

′ 12 −1 ′ 12 21 − 12 √ log t max ′ √

≪q x t + q x t + e (f (b))

16N6t−a n N N6b6N+h62N

16n6x √ X 1 1 1 1 1 1 q′ 1 √ log t max ′ √ t 6 N 2 ≪ q ′ x 2 t−1 + q ′ 2 x 2 t− 2 + 16N6t−a n N 16n6x 1

1

1

≪ q ′ x 2 t−1 + q ′ 2 x 2 t− 2 + q ′ 2 x 2 t 6 log t 1

≪ q ′ x 2 t−1 + q ′ 2 x 2 t 6 log t.

(5-32)

aq/, Â?ÂŒÂą M2 O, 1

1

1

1

M2 ≪ q ′ y 2 t−1 + q ′ 2 y 2 t 6 log t. 118

(5-33)

1ÊÙ Ù§êدK

Ïd, dª(5-24), (5-32)Ú(5-33), k

!

′

X ′

q

1 1 b+a 1

ζ − it, −

′ b+a′ 12 +it b+a′ 12 −it

2 q ( q′ )

b=1 ( q′ )

1 1 1 1 1 1 1 ≪ q ′ x 2 + y 2 t−1 + q ′ 2 x 2 + y 2 t 6 log t + q ′ x− 2 log t.

q t x = y = 2π ,

!

q′ ′ X

1 1 b+a 1

≪ q ′ 12 t 125 log t + q ′ t− 14 log t. ζ − − it, 1 1 ′ ′

′ +it −it

2 q 2 2 ( b+a

b=1 ( b+a

q′ ) q′ )

ùÒy² Ún5.18.

Ú n 5.19 q ′ > 2 ´ ê, 0 6 a′ 6 1, t > 3. Kk

! 2 q′

X 1 1 b + a′

+ it, − b+a′ 1 +it

ζ ′

2 q ( q′ ) 2 b=1 1 1 5 q′ t ′ − 12 3 ′2 6 log t) + O ) + O(t log t = q ′ log ′ + γ + O(q t q , (q + a′ )2π Ù¥γ ´Euler~ê, O ~ê= 6ua′ .

y²: â5 5.7, |^)Ûòÿk

! 2 q′

X 1 1 b + a′

+ it, −

ζ

1 ′ ′ b+a 2 +it

2 q ( ) ′ b=1 q

2

b+a′ q′

it

X X X e − q′ v π 1 1 2π

i(t+ 4 ) −2 = + e + O(x log t)

1 1 +it −it

t ′ 2 v2 16v6y b=1 16n6x n + b+a

q′

2

2 ′

′

′

q q b+a X X X X e(− q′ v)

1

+

+ = 1

b+a′ 21 +it

−it 2 (n + v )

b=1 16n6x b=1 16v6y q′     −it q′ b+a′ X X X e( q′ v) 1 −i(t+ π4 )    2π + + e 1 1 ′ b+a 2 +it +it t 2 (n + ) v ′ 16n6x 16v6y b=1 q    ′ it q b+a′ X X X e(− v) π 1 q′    2π + + ei(t+ 4 ) 1 b+a′ 12 −it −it t 2 v 16n6x (n + q ′ ) 16v6y b=1

 

it b+a′ X X X e(− v)

′ 1 π 1 2π q − i(t+ )

 + 4 +O x 2 log t + e 1 1

b+a′ 2 +it −it t 2 (n + ) v

16b6q ′ 16n6x 16v6y q′ +O x−1 log2 t ≡ A1 + A2 + A3 + A4 + A5 + O x−1 log2 t . (5-34) 119

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

ò OÂŞ(5-34)ÂĽz˜‘. (i)Ă„k OA1 . dÂŞ(5-34) k

2

q′

X X

1

A1 = 1 ′

b+a 2 +it

(n + )

b=1 16n6x q′ ′

=

q X X

1

X

b=1 16n6x 16m6x ′

q X X

(n +

b+a′ 12 +it ) q′

1 b+a′ 12 −it ) q′

(m +

′

q X X 1 1 1 = + + ′ 1 +it ′ 1 −it b+a′ b+a b+a n + 2 2 (n + ) (m + ) ′ ′ ′ q b=1 16n6x b=1 16m<n6x q q ′

+

q X

1

X

b=1 16n<m6x q′

= q′

b=1 16n6x

b+a′ 12 −it q′ )

X

1

b=1 16m<n6x

′

+

(m +

q′

X 1 + nq ′ + b + a′

X X q X

(n +

1

b+a′ 12 +it q′ )

1

X

b=1 16m<n6x

(n +

b+a′ q′

(n +

1

b+a′ q′

)

1 +it 2

(m +

b+a′ 12 −it q′ )

1 )

1 −it 2

(m +

b+a′ 12 +it ) q′

= q ′ A11 + A12 + A13 .

(5-35)

w,, A13 †A12 �. �d�I OA11 ÚA12 . ÊuA11 , �� q X

q q ∞ X X 1 1 X a a = − = log q + Îł − + O(q −1 ), n + a n n(n + a) n(n + a) n=1 n=1 n=1 n=1

k ′

A11 =

q X X

b=1 16n6x

nq ′

1 + b + a′

X 1 1 − ′ n+a n + a′ 16n6q ′ 16n6q ′ ([x]+1) 1 1 ′ ′ = log(q ([x] + 1)) + Îł − βa′ + O − log q + Îł − βa′ + O ′ qx q′ 1 = log x + O ′ , (5-36) q P a′ Ă™¼βa′ = ∞ n=1 n(n+a′ ) .

=

X

eÂĄ OA12 . Ă„kk

′

A12 =

q X

X

b=1 16m<n6x

(n +

120

m+ b+a q′ n+ b+a q

′

′

b+a′ 21 q ′ ) (m

it

+

b+a′ 12 q′ )

1ÊÙ Ù§êدK ′

=

q X

X

16m<n6x b=1

it log

e (n +

mq ′ +b+a′ nq ′ +b+a′

b+a′ 21 ) (m q′

+

b+a′ 12 ) q′

.

(5-37)

f (b) =

mq ′ + b + a′ t log ′ , 2π nq + b + a′

(5-38)

1 1 b + a′ − 2 b + a′ − 2 , g(b) = n + m+ q′ q′

(5-39)

w,, g(b) ´üNeü , (n − m)q ′ t > 0, 2π (mq ′ + b + a′ )(nq ′ + b + a′ )

(5-40)

−t(n − m)q ′ ((n + m)q ′ + 2(b + a′ )) < 0, 2π(mq ′ + b + a′ )2 (nq ′ + b + a′ )2

(5-41)

q′ q ′ (m + a′ ) + 2(b + a′ ) . 2 (nq ′ + b + a′ ) 32 (mq ′ + b + a′ ) 32

(5-42)

f ′ (b) =

f ′′ (b) =

g ′ (b) = −

Ïd, f (b) ´üNþ, , f ′ (b) ´üNeü . t 6

mnq ′ , n−m

k |f ′ (b)| =

t (n − m)q ′ 1 6 < 1, ′ ′ ′ ′ 2π (mq + b + a )(nq + b + a ) 2π

|g(b)| = |g ′ (b)| =

n+

b+a′ q′

1 1 2

m+

1 1 6 √mn b+a′ 2

1 6√ , mn 2 (nq ′ + b + a′ ) (mq ′ + b + a′ ) 3 2

f ′ (b) t(n − m) (n − m)t √ = . 1 1 ≫ g(b) mnq ′ 2π(mq ′ + b + a′ ) 2 (nq ′ + b + a′ ) 2 dd9 â5 5.3Úª(5-35), (5-43)-(5-46),   X  1 + √1  |A12 | ≪ (n−m)t mn √ 16m<n6x mnq ′ √ X mn ′ −1 ≪q t +x n−m 16m<n6x

121

(5-44)

q′

q ′ (q ′ (m + n) + 2(b + a′ )) 3 2

(5-43)

(5-45)

(5-46)

Smarandache ¼ê9Ù '¯KïÄ

′ −1

=qt

x−1 X x−1 X

k=1 m=1 ′ −1 2

p

m(m + k) +x k

≪ q t x log t. t >

mnq ′ n−m ,

(5-47)

dª(5-41), k t(n − m) t(n − m) ≪ |f ′′ (b)| ≪ , 2 ′2 m q n m2 q ′2 n

Ïdd5 5.4 Úª(5-37), k

′ +b+a′

X log q

it log mq nq ′ +b+a′ max ′

e |A12 | ≪

1 1 16N6q

′ 2 ′ 2 1+a 1+a 16m<n6x n + N<b<N +h<2N m + q′ q′ X log t ≪ 1 1 × ′ 2 ′ 2 1+a 1+a 16m<n6x n + m + q′ q′

1 − 1

2 2 t(n − m) t(n − m)

× max ′ q ′ +

2 ′2 2 ′2 16v6q

m q n m q n √ √ X X t log t t log t √ ≪ √ √ 3+ nm n m X

16m<n6x 1 2

16m<n6x

′ 2 −1

≪ t x log t + q x t

log t.

(5-48)

¤±, dª(5-37), (5-47)Ú(5-48), k 1 ′ −1 2 2 |A12 | ≪ q t x + t x log t.

Ïd, dª(5-35), (5-36)Ú(5-49), 1 A1 = q ′ log x + O (q ′ t−1 x2 + t 2 x) log t .

Ó , |^CauchyØ ª, k

2  12 1  ′

′

′ 2

q q q X X X X X

1 1 2

6

   1 1 1

b+a′ 2 +it

b+a′ 2 +it

) ) (n + (n +

b=1 16n6x b=1 b=1 16n6x q′ q′ 1

≪ q ′ log 2 t.

(ii)éuA2 , Ï

P

16u6y

1 u

= log y + γ + O q1′ , k

2 ′

q′

X v)

X e(− b+a

q′

A2 = 1

−it 2 v

b=1 16v6y

122

(5-49)

(5-50)

(5-51)

1ĂŠĂ™ Ă™§êĂ˜ÂŻK ′

=

X

q 2Ď€i X X e

b+a′ (v−u) q′

1

1

u 2 −it v 2 +it   X 1 X 1   = q′ + O q ′ 1 1  u u2 v 2 16u<v6y 16u6y u≥v( mod q ′ )  X = q ′ (log y + Îł + O(q â€˛âˆ’1 )) + O q ′ 16u6y 16v6y b=1

X

16k6[ qy′ ] 16u6y



3

= q ′ (log y + Îł) + O q ′ 4

1

X

16k6[ qy′ ]

= q ′ (log y + γ) + O(y).

k

1 4

X

16u6y



1  3 u4



1  √ √ ′ u kq + u

(5-52)

|^CauchyĂ˜ ÂŞ, ĂĄ=



2  12 1  ′

′ ′ 2

q′

q′ q

b+a b+a X X e(− q′ v)

X X X e(− q′ v)

2 

  1 1 1

6

−it −it 2 2 v v

b=1 16v6y b=1 b=1 16v6y 1

≪ q ′ log 2 t.

(5-53)

(iii) x < y, 2Ď€xy = t, KŠâ5Â&#x;5.7, k ′ it 1 X X e(− b+a v) 1 2Ď€ q′ i(t+ Ď€4 ) −2 + e + O x log t 1 b+a′ 21 +it t v 2 −it 06n6x (n + q ′ ) 16v6y ′ it 1 X X e(− b+a ′ v) Ď€ 1 2Ď€ q i(t+ 4 ) −4 = + e + O t . log t 1 ′ 1 +it −it t 2 2 √ t (n + b+a √ ) v ′ t q 06n6

16v6

2Ď€

2Ď€

Ă?d,

2Ď€ t

it

2Ď€ = t

it

+

′

i(t+ π4 )

e

1

v 2 −it

16v6y i(t+ π4 )

e

X

√

x<n6

X e(− b+a q ′ v) X

√

16v6

1

t 2Ď€

1 t 2Ď€

(n +

′

e(− b+a v) q′

b+a′ 21 +it q′ )

v 2 −it 1 −2 + O x log t .

dA3 ½Ă‚Úª(5-54), (5-51) ĂĄ=  ′ −it q X X 1  2Ď€ −i(t+ Ď€4 ) A3 = e  1 ′ b+a 2 +it t b=1 16n6x (n + q ′ ) 123

(5-54)

′

X

√

16v6

e( b+a q ′ v) 1

t 2Ď€

v 2 +it

+

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

X

+

√

x6n6

2Ď€ = t

−it

1 t 2Ď€

b+a′ 12 −it q′ )

(n +

−i(t+ Ď€4 )

e

X



1  + O x− 2 log t 

√

16n6x 16v6

1

+

X

√

16n6x x6m6

t 2Ď€

q X b=1

1 +O q ′ 2 log t

n+

v 2 +it

t 2Ď€

′

X

′

1

X

b+a′ q′

1 1 +it 2

q X b=1

′

e( b+a v) q′ 1 +it + b+a′ 2 n + q′ b+a′ q′

m+

1 −it + 2

1 = A31 + A32 + O q ′ 2 log t .

(5-55)

ĂŠuA32 , ÂŒÂą|^† OA1 aq Â?{ , Ă?dk 3

3

3

1

|A32 | ≪ q ′ x 2 t− 4 + t 4 x 2 log t.

(5-56)

ĂŠuA31 , dÂŞ(5-55), k |A31 | 6

6

X

X

√

16n6x 16v6

X

X

√

16n6x 16v6

-

1 t 2Ď€

t 2Ď€

v

1 +it 2

′

′

2πi b+a v q′

q e X b=1

n+

n+

b+a′ q′

w,, g(b) ´ßNeĂź , {ĂźOÂŽ v t − , ′ ′ q 2Ď€(nq + b + a′ )

f ′′ (b) =

1 2

−it

q′ b+a′ b+a′ 2Ď€i v−it log n+

q′ q′ 1

X e

. 1 1

v 2

b=1 b+a′ 2

n + q′

t b + a′ b + a′ v− , f (b) = log n + q′ 2Ď€ q′ 1 b + a′ − 2 g(b) = n + , q′

f ′ (b) =

b+a′ q′

2π(nq ′

g ′ (b) = −

t , + b + a′ )2

1 2q ′ n +

124

b+a′ q′

3 . 2

(5-57)

1ÊÙ Ù§êدK

Ïd, t 6 nq ′ ,

v

t t t 1 ′

6 |f (b)| = ′ − 6 6 < 1,

′ ′ ′ ′ ′ q 2π(nq + b + a ) 2π(nq + b + a ) 2πnq 2π

1

′ −2

b+a 1

|g(b)| 6 n +

6 √ , ′

q n

(5-58)

(5-59)

1 |g ′(b)| 6 √ , n

(5-60)

f ′ (b) − = g(b)

v t − ′ ′ ′ 2π(nq + b + a ) q

v t > − ′ ′ ′ 2π(nq + b + a ) q t t √ ≫ ′ n ≫ ′√ . nq q n

b + a′ n+ q′ 1 + a′ n+ q′

1 2

1 2

(5-61)

Ïd, d5 5.3 Úª(5-57)-(5-61), k |A31 | ≪

X

X

√

16n6x 16v6

≪ q ′ t−1

1

t √ q′ n

1

t 2π

X √ n

16n6x

3

1

v2

X

√

16v6

t 2π

1 +√ n

!

1 √ v

3

≪ q ′ t− 4 x 2 . t > nq ′ , k n2tq′2 ≪ |f ′′ (b)| ≪ |A31 | ≪ ≪ ≪

X

X

t 2π

X

X

t 2π

X

X

√

16n6x 16v6

√

16n6x 16v6

√

16n6x 16v6 3

t 2π

1

(5-62) t , n2 q ′2

¤± â5 5.4 Úª(5-57),

X 1 log q ′

2πif (b)

√ max e

1

n 1<N<q′

v2 N<b<N +h<2N " 1 2 ′2 12 # 2 1 log q ′ ′ t n q √ q + 1 2 ′2 n n q t v2 log t √ nv 1

√ t nq ′ +√ n t

3

≪ t 4 x 2 log t + q ′ t− 4 x 2 log t.

(5-63)

(ܪ(5-62)Ú(5-63), k 3

3

3

1

1

3

|A31 | ≪ q ′ t− 4 x 2 + t 4 x 2 log t + q ′ t− 4 x 2 log t 125

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„ 3

3

1

1

≪ q ′ x 2 t− 4 log t + t 4 x 2 log t.

(5-64)

Ă?d, dÂŞ(5-55), (5-56)Ăš(5-64), k 3

3

3

1

1

|A3 | ≪ q ′ x 2 t− 4 + t 4 x 2 log t + q ′ 2 log t.

(5-65)

A4 †A3 ´ Ă? , ¤¹kaq O, = 3

3

3

1

1

|A4 | ≪ q ′ x 2 t− 4 + t 4 x 2 log t + q ′ 2 log t.

(5-66)

(iv)ĂšĂœŠò OA5 . dÂŞ(5-34), (5-51)Ăš(5-53), ĂĄ=

′ b+a X

X 1 1 2Ď€ it i(t+ Ď€ ) X e − q′ v

4 |A5 | ≪ x− 2 log t + e

1 1 +it −it

t ′ 2 2 v b+a ′ 16b6q 16n6x n + 16v6y

q′

 

b+a′

e − v X X

 q′ 1  X X

− 12



6 x log t  +

1 1

+it

−it

′ 2 2 v

b+a ′ ′ 16b6q 16n6x n + 16b6q 16v6y

q′ 1

3

6 q ′ x− 2 log 2 t.

nÞ¤ã, (Ăœ(i), (ii), (iii)Ăš(iv)oĂœŠ y², ĂĄ= 1 A = q ′ log x + q ′ (log y + Îł) + O q ′ t−1 x 2 log t + 1 3 3 1 +O t 2 x log t + O(y) + O q ′ x 2 t− 4 + O q ′ 2 log t 3 1 3 ′ − 12 4 2 2 +O t x log t + O q x log t + O x−1 log2 t . 1

-x = t 6 log3 t, y =

t 2Ď€x ,

A = q ′ log ÚÒy² Ăšn5.19.

(5-67)

k

5 1 1 5 t + q ′ Îł + O q ′ t− 12 + O t 6 log 2 t + O q ′ 2 log t . 2Ď€

ĂšÂ˜!, ò ¤½n5.5 y². ŠâĂšn5.17ĂŠN´

2 X 1

L

+ it, χ, a

2 χ mod q

q′ X

φ(q) X 1 b + a′

2

= Âľ(d)

Μ 2 + it, q ′

q b=1 d|q

! 2 q′

′ X φ(q) X 1 1 b + a 1

+ it, = Âľ(d) −

b+a′ 1 +it + Μ

b+a′ 21 +it

( ′ )2 q 2 q′ ( ) ′ b=1 q q d|q

126

(5-68)

1ĂŠĂ™ Ă™§êĂ˜ÂŻK q′

X q′ φ(q) X = ¾(d) q b + a′ d|q

b=1

q′

X φ(q) X 1 + ¾(d) 1 ′ q ( b+a′ ) 2 +it d|q

b=1

q

Îś

b + a′ 1 − it, 2 q′

−

1 ′

1

−it 2 ( b+a q′ )

!

+

! q X φ(q) X 1 1 b + a′ 1 + + it, Âľ(d) Îś − b+a′ 1 + b+a′ 12 −it q 2 q′ ( q′ ) 2 +it b=1 ( q ′ ) d|q

! 2 q′

X 1 b + a′ φ(q) X 1

+ Âľ(d) + it, − b+a′ 1

Îś

. ′ +it

q 2 q ( ′ )2 ′

d|q

b=1

q

2dĂšn5.18, 5.19Ăš5.16, ĂŠ?¿¢êt > 3, k 2 X

1

L + it, χ, a

2 χ mod q

φ(q) X Âľ(d)q ′ log(q ′ ) + Îł − βa′ + O q â€˛âˆ’1 + q d|q 1 5 1 t φ(q) X ′ ′ − 12 3 ′ Âľ(d) q log + Îł + O(q t + + t 6 log t + q 2 log t) + q 2Ď€ d|q   X 1 5 1 φ(q) +O  |Âľ(d)|(q ′ 2 t 12 log t + q ′ t− 4 log t) q d|q X Âľ(d) qt X Âľ(d) X Âľ(d) log d = φ(q) log + 2Îł − φ(q) βa/d − φ(q) + d 2Ď€ d d d|q d|q d|q   X X X 1 |Âľ(d)| φ(q) 5 φ(q) 5 |Âľ(d)|  +O φ(q)t− 12 + t 6 log3 t |Âľ(d)| + 1 t 12 log t 1 d q q2 d2 d|q d|q d|q   X log p X Âľ(d) φ2 (q)  qt  − φ(q) = log + 2Îł + βa/d q 2Ď€ p−1 d p|q d|q 1 5 1 5 − 12 3 ω(q) ω(q) 6 2 12 +O qt + t log t2 + q t log t2 ,

=

P∞

x n=1 n(n+x)

´Â˜Â‡Â†x k' ÂŒOÂŽ ~ĂŞ, …ω(q) LÂŤq Ă˜Ă“ÂƒĂ?f P q ‡ê. Þª¼|^ OÂŞ d|q |Âľ(d)| ≪ φ(q) . ÚÒ ¤ ½n5.5 y². d Ă™¼βx =

127

Smarandache Ÿê9Ă™Âƒ'ÂŻKĂŻĂ„

Í Šz

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