A new additive function and the F. Smarandache function

Scientia Magna Vol. 5 (2009), No. 1, 128-132

A new additive function and the F. Smarandache function Yanchun Guo Department of Mathematics, Xianyang Normal University Xianyang, Shaanxi, P.R.China Abstract For any positive integer n, we define the arithmetical function F (n) as F (1) = 0. αk 1 α2 If n > 1 and n = pα be the prime power factorization of n, then F (n) = α1 p1 + 1 p2 · · · pk α2 p2 + · · · + αk pk . Let S(n) be the Smarandache function. The main purpose of this paper is using the elementary method and the prime distribution theory to study the mean value properties of (F (n) − S(n))2 , and give a sharper asymptotic formula for it.

Keywords Additive function, Smarandache function, Mean square value, Elementary method, Asymptotic formula.

§1. Introduction and result Let f (n) be an arithmetical function, we call f (n) as an additive function, if for any positive integers m, n with (m, n) = 1, we have f (mn) = f (m) + f (n). We call f (n) as a complete additive function, if for any positive integers r and s, f (rs) = f (r) + f (s). In elementary number theory, there are many arithmetical functions satisfying the additive properties. For αk 1 α2 example, if n = pα 1 p2 · · · pk denotes the prime power factorization of n, then function Ω(n) = α1 + α2 + · · · + αk and logarithmic function f (n) = ln n are two complete additive functions, ω(n) = k is an additive function, but not a complete additive function. About the properties of the additive functions, one can find them in references [1], [2] and [5].

In this paper, we define a new additive function F (n) as follows: F (1) = 0; If n > 1 and n = αk 1 α2 pα p 1 2 · · · pk denotes the prime power factorization of n, then F (n) = α1 p1 +α2 p2 +· · ·+αk pk . αk 1 α2 It is clear that this function is a complete additive function. In fact if m = pα 1 p2 · · · pk αk +βk βk α1 +β1 α2 +β2 β1 β2 · · · pk . Therefore, F (mn) = p2 and n = p1 p2 · · · pk , then we have mn = p1 (α1 + β1 )p1 + (α2 + β2 )p2 + · · · + (αk + βk )pk = F (m) + F (n). So F (n) is a complete additive function. Now we let S(n) be the Smarandache function. That is, S(n) denotes the smallest positive integer m such that n divide m!, or S(n) = min{m : n | m!}. About the properties of S(n), many authors had studied it, and obtained a series results, see references [7], [8] and [9]. The main purpose of this paper is using the elementary method and the prime distribution theory to study the mean value properties of (F (n) − S(n))2 , and give a sharper asymptotic formula for it. That is, we shall prove the following: Theorem.

Let N be any fixed positive integer. Then for any real number x > 1, we

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A new additive function and the F. Smarandache function

have the asymptotic formula X

2

(F (n) − S(n)) =

N X

ci ·

i=1

n≤x

x2 lni+1 x

µ +O

x2 lnN +2

where ci (i = 1, 2, · · · , N ) are computable constants, and c1 =

¶ √

x

,

π2 6 .

§2. Proof of the theorem In this section, we use the elementary method and the prime distribution theory to complete the proof of the theorem. We using the idea in reference [4]. First we define four sets A, B, C, D as follows: A = {n, n ∈ N , n has only one prime divisor p such that p | n and p2 - n, 1 1 p > n 3 }; B = {n, n ∈ N , n has only one prime divisor p such that p2 | n and p > n 3 }; 1 C = {n, n ∈ N , n has two deferent prime divisors p1 and p2 such that p1 p2 | n, p2 > p1 > n 3 }; 1 D = {n, n ∈ N , any prime divisor p of n satisfying p ≤ n 3 }, where N denotes the set of all positive integers. It is clear that from the definitions of A, B, C and D we have X

2

(F (n) − S(n))

X

=

n≤x

2

(F (n) − S(n)) +

X

2

(F (n) − S(n))

n≤x n∈B

n≤x n∈A

+

X

2

(F (n) − S(n)) +

n≤x n∈C

X

2

(F (n) − S(n))

n≤x n∈D

≡ W1 + W2 + W3 + W4 .

(1)

Now we estimate W1 , W2 , W3 and W4 in (1) respectively. Note that F (n) is a complete additive function, and if n ∈ A with n = pk, then S(n) = S(p) = p, and any prime divisor q of 1 1 k satisfying q ≤ n 3 , so F (k) ≤ n 3 ln n. From the Prime Theorem (See Chapter 3, Theorem 2 of [3]) we know that k X

X

x π(x) = 1= ci · i + O ln x i=1 p≤x

µ

x lnk+1 x

¶ ,

(2)

where ci (i = 1, 2, · · · , k) are computable constants, and c1 = 1. By these we have the estimate: W1

=

X

2

(F (n) − S(n)) =

n≤x n∈A

=

X

F 2 (k) ¿

pk≤x (pk)∈A 2

¿ (ln x)

X √ k≤ x

2

k3

X

X pk≤x (pk)∈A

X

√ x k≤ x k<p≤ k

³ x ´ 53 k

2

(F (pk) − p)

2

2

(pk) 3 ln2 (pk) ≤ (ln x)

5 1 ¿ x 3 ln2 x. ln xk

X √ k≤ x

2

k3

X

2

p3

k<p≤ x k

(3)

130

Yanchun Guo

No. 1

If n ∈ B, then n = p2 k, and note that S(n) = S(p2 ) = 2p, we have the estimate X ¡ X ¢2 2 (F (n) − S(n)) = F (p2 k) − 2p W2 = n≤x n∈B

X

=

X

1 k≤x 3

√x

k<p≤

X k2 · x

¿

p2 k≤x p>k

k 2 ln x

1

X

1 k≤x 3

k

1 2

1

1 k≤x 3

F 2 (k) ¿

X √x

k<p≤

k2

k

4 3

x . ln x

¿

(4)

1

If n ∈ D, then F (n) ≤ n 3 ln n and S(n) ≤ n 3 ln n, so we have X X 2 5 2 W4 = (F (n) − S(n)) ¿ n 3 ln2 n ¿ x 3 ln2 x. n≤x n∈D

(5)

n≤x 1

Finally, we estimate main term W3 . Note that n ∈ C, n = p1 p2 k, p2 > p1 > n 3 > k. If 1 k < p1 < n 3 , then in this case, the estimate is exact same as in the estimate of W1 . If 1 k < p1 < p2 < n 3 , in this case, the estimate is exact same as in the estimate of W4 . So by (2) we have ³ 5 ´ X X 2 2 W3 = (F (n) − S(n)) = (F (p1 p2 k) − p2 ) + O x 3 ln2 x n≤x n∈C

=

X

X ¡

X √x p

1

k≤x 3 k<p1 ≤

=

p1 p2 k≤x p2 >p1 >k

X

k

x 2≤ p k 1

X

X

√x p

1

k≤x 3 k<p1 ≤

³ 5 ´ ¢ F 2 (k) + 2p1 F (k) + p21 + O x 3 ln2 x

k





X p21 + O 

x 1 <p2 ≤ p k 1

X

1

X

√x p

k≤x 3 k<p1 ≤

x 1 <p2 ≤ p k 1

k

µ ¶! ³ 5 ´ x x 2 3 ln x + O = ci · + O x √x p1 k lni px1 k p1 k lnN +1 x 1 i=1 k≤x 3 k<p1 ≤ k   X X X X X X   − p21 1+O kp1  . √ x p2 ≤p1 √ x p <p ≤ x 1 1 X

X

k≤x 3

k<p1 ≤

p21

ÃN X

³ 5 ´  kp1  + O x 3 ln2 x

k≤x 3

k

k<p1 ≤

k

1

2

(6)

p1 k

2

π 6

, from the Abel’s identity (See Theorem 4.2 of [6]) and (2) we have "N ¶# µ X X X ci · p1 X X X p1 2 2 1= p1 p1 +O √ x p≤p1 √x lni p1 lnN +1 p1 1 1 i=1 k<p ≤ k<p ≤ 1 1 k≤x 3 k≤x 3 k k   N X 3 3 X X X X ci · p1 p1   = +O  i N +1 √ √ ln p ln p 1 1 1 1 x x i=1

Note that ζ(2) =

k≤x 3 k<p1 ≤

=

N X i=1

di · x2 +O lni+1 x

k≤x 3 k<p1 ≤

k

µ

2N · x2 lnN +2 x

k

¶ ,

(7)

Vol. 5

where di (i = 1, 2, · · · , N ) are computable constants, and d1 = X 1 k≤x 3

131

A new additive function and the F. Smarandache function

X

X

√x p

k<p1 ≤

x 1 <p2 ≤ p k 1

k

X

X

kp1 ¿

X

k

1 k≤x 3

√x

p1 ≤

p1 ·

π2 6 .

5 3 X x3 x x2 √ ¿ ¿ . p1 k ln x ln2 x k ln2 x 1

k≤x 3

k

X p1 x x2 x2 ¿ ¿ .

N +2 2 √ x k lnN +1 x x lnN +2 x 1 k ln

X

1 k≤x 3

k<p1 ≤

(8)

(9)

k≤x 3

k

From the Abel’s identity and (2) we also have the estimate X

X √x

1

k≤x 3 k<p1 ≤

=

N X i=1

bi ·

p21

X 1 x x = p1 k ln p1 k k 1 k≤x 3

k

µ

x2 i+1

ln

x

+O

ln

xp1 x √ x ln kp1

k<p1 ≤

k

¶

x2 N +1

X

x

(10)

, 2

where bi (i = 1, 2, · · · , N ) are computable constants, and b1 = π3 .

Now combining (1), (3), (4), (5), (6), (7), (8)and(9) we may immediately deduce the asymptotic formula: X n≤x

2

(F (n) − S(n)) =

N X i=1

ai ·

x2 lni+1 x

µ +O

x2 lnN +2

¶ √

x

,

where ai (i = 1, 2, · · · , N ) are computable constants, and a1 = b1 − d1 = This completes the proof of Theorem.

π2 6 .

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[2] H.N.Shapiro, Introduction to the theory of numbers, John Wiley and Sons, 1983.

[3] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem (in Chinese), Shanghai Science and Technology Press, Shanghai, 1988.

[4] Xu Zhefeng, On the value distribution of the Smarandache function, Acta Mathematica Sinica (in Chinese), 49(2006), No.5, 1009-1012.

[5] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi’an, 2007.

[6] Tom M. Apostol. Introduction to Analytic Number Theory, Springer-Verlag, 1976.

[7] Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems (in Chinese), High American Press, 2006.

[8] Chen Guohui, New Progress On Smarandache Problems (in Chinese), High American Press, 2007.

[9] Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress (in Chinese), High American Press, 2008.