Some arithmetical properties of the Smarandache series

Scientia Magna Vol. 10 (2014), No. 1, 64-67

Some arithmetical properties of the Smarandache series Qianli Yang Department of Mathematics, Weinan Normal University, Weinan, China 714000 Abstract The Smarandache function S(n) is defined as the minimal positive integer m such that n|m!. The main purpose of this paper is to study the analyze converges questions for ∞ ∞ P P 1 1 , i.e., we proved the series diverges for any δ ≤ 1, some series of the form S(n)δ S(n)δ and

∞ P n=1

n=1

n=1

1 S(n) S(n)

converges for any > 0.

Keywords Smarandache function, smarandache series, converges.

§1. Introduction and results For every positive integer n, let S(n) be the minimal positive integer m such that n|m!, i.e., S(n) = min{m : m ∈ N, n|m!}.

This function is known as Smarandache function [1] . Easily, one has 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, ¡ ¡ ¡ . Îąk 1 Îą2 Use the standard factorization of n = pÎą 1 p2 ¡ ¡ ¡ pk , p1 < p2 < ¡ ¡ ¡ < pk , itâ&#x20AC;&#x2122;s trivial to have i S(n) = max {S(pÎą i )}.

1â&#x2030;¤iâ&#x2030;¤k

[2]

Many scholars have studied the properties of S(n), for example, M. Farris and P. Mitchell show the boundary of S(pÎą ) as (p â&#x2C6;&#x2019; 1)Îą + 1 â&#x2030;¤ S(pÎą ) â&#x2030;¤ (p â&#x2C6;&#x2019; 1)[Îą + 1 + logp Îą] + 1.

Z. Xu

[3]

noticed the following interesting relationship formula Ď&#x20AC;(x) = â&#x2C6;&#x2019;1 +

[x] X S(n) n=2

n

,

by the fact that S(p) = p for p prime and S(n) < n except for the case n = 4 and n = p, where Ď&#x20AC;(x) denotes the number of prime up to x, and [x] the greatest integer less or equal to x. Those and many other interesting results on Smarandache function S(n), readers may refer to [2]-[6].

1 This

work is supported by by the Science Foundation of Shanxi province (2013JM1016).

Vol. 10

65

Some arithmetical properties of the Smarandache series

Let p be a fixed prime and n â&#x2C6;&#x2C6; N, the primitive numbers of power p, denoted by Sp (n), is defined by Sp (n) = min{m : m â&#x2C6;&#x2C6; N, pn |m!} = S(pn ). â&#x2C6;&#x17E; P 1 Z. Xu [3] obtained the identity between Riemann zeta function Îś(s) = ns , Ď&#x192; > 1 and an n=1

infinite series involving Sp (n) as

â&#x2C6;&#x17E; X

1

S s (n) n=1 p

=

Îś(s) , ps â&#x2C6;&#x2019; 1

and he also obtained some other asymptotic formulae for Sp (n). F. Luca [4] proved the series â&#x2C6;&#x17E; â&#x2C6;&#x17E; P P 1 1 converges for all δ â&#x2030;Ľ 1 and diverges for all δ < 1, and the series S(n)δ S(n) log n n=1

S(n)

n=1

converges for any Îľ > 0.

In this note, we studied the analyze converges problems for the infinite series involving S(n). That is, we shall prove the following conclusions: Theorem 1.1. For any δ â&#x2030;¤ 1, the series â&#x2C6;&#x17E; X

1 S(n)δ n=1 diverges.

Theorem 1.2. For any Îľ > 0, the series â&#x2C6;&#x17E; X

1 ÎľS(n) S(n) n=1 converges.

§2. Some lemmas To complete the proof of theorems, we need two Lemmas.

Lemma 2.1. Let p be any fixed prime. Then for any real number x â&#x2030;Ľ 1, we have the asymptotic formula:

â&#x2C6;&#x17E; X 1 1 p ln p 1 = ln x + Îł + + O(xâ&#x2C6;&#x2019; 2 + ), Sp (n) pâ&#x2C6;&#x2019;1 pâ&#x2C6;&#x2019;1 n=1 Sp (n)â&#x2030;¤x

where Îł is the Euler constant, denotes any fixed positive numbers.

Proof. See Theorem 2 of [3].

Lemma 2.2.[7] Let > 0 and d(n) denotes the divisor function of positive integer n.

Then d(n) = O(n ) â&#x2030;¤ C n , where the o-constant C depends on .

Q Îą Proof. The proof follows [7] by writing n = p , the standard factorization of n. Then p|n

pÎą â&#x2030;Ľ 2Îą = eÎą ln 2 â&#x2030;Ľ Îą ln 2 â&#x2030;Ľ

1 (a + 1) ln 2.

2

66

Qianli Yang

No. 1

If p â&#x2030;Ľ 2, then pÎą â&#x2030;Ľ 2Îą â&#x2030;Ľ Îą + 1. Therefore, Y Îą+1 Y Îą+1 Y d(n) Y Îą + 1 = = â&#x2030;Ľ

Îą Îą Îą n p p p p|n

p|n p <2

The last item in above inequality is

p|n p <2

p|n p â&#x2030;Ľ2

Q p|n p <2

2 ln 2 ,

1 2 (a

Y Îą+1 Îą+1 . Îą+1 + 1) ln 2 p|n p â&#x2030;Ľ2

Q

which is less than

p <2

2 ln 2

= C , say, the

o-constant C depends on .

§3. Proof of theorems Proof of Theorem 1.

We may treat the case δ = 1 first. By Lemma 1 and the notation Sp (n) = S(pn ), we have â&#x2C6;&#x17E; X

â&#x2C6;&#x17E; X 1 = lim S(pn ) xâ&#x2020;&#x2019;+â&#x2C6;&#x17E; n=1 n=1

1 = â&#x2C6;&#x17E;.

Sp (n)

Sp (n)â&#x2030;¤x

Obviously, for δ â&#x2030;¤ 1,

â&#x2C6;&#x17E; P n=1

1 S(n)δ

diverges follows easily by the trivial inequality:

â&#x2C6;&#x17E; X

â&#x2C6;&#x17E; â&#x2C6;&#x17E; X X 1 1 1 â&#x2030;Ľ â&#x2030;Ľ , δ S(n) S(n) n=1 S(pn ) n=1 n=1

complete the proof.

Proof of Theorem 2. â&#x2C6;&#x17E; P It certainly suffices to assume that â&#x2030;¤ 1. We rewrite series n=1 â&#x2C6;&#x17E; X u(k) k=1

k Îľk

1 S(n)ÎľS(n)

as

,

where u(k) = ]{n : S(n) = k}. For every positive integer n such that S(n) = k is a divisor of k!, i.e. u(k) â&#x2030;¤ d(k!). By Lemma 2 and the inequality bellow 2

(k!) =

k Y j=1

j(k + 1 â&#x2C6;&#x2019; j) <

2 k Y k+1 j=1

2

=

we have

u(k) â&#x2030;¤ d(k!) â&#x2030;¤ C (k!) < C

k+1 2

k+1 2

2k .

k .

where C means that the constant depending on . k Therefore, recalling that the properties of the sequence 1 + k1 , we have â&#x2C6;&#x17E; X u(k) k=1

k Îľk

â&#x2030;¤ C

k

k â&#x2C6;&#x17E; â&#x2C6;&#x17E; â&#x2C6;&#x17E; X X X 1 k+1 1 k+1 1 = C < C ,

1 k Îľk 2 2Îľk k 2Îľk

k=1

k=1

k=1

Vol. 10

Some arithmetical properties of the Smarandache series

for some constant C1 , it follows that series

∞ P k=1

C1

u(k) kεk

67

is bounded above by

∞ X C1 1 = ε , 2εk 2 −1

k=1

completing the proof.

References [1] F. Smarandache, Only problem not solution, Xiquan Publishing House, Chicago, 1993.

[2] M. Farris and P. Mitchell, Bounding the Smarandache function, Smarandache Notation Journal, 13(2002), 37-42.

[3] Z. Xu, Some arithmetical properties of primitive numbers of power p, 2(2006), No. 1, 9-12.

[4] F. Luca, The Smarandache harmonic series, NAW, 5(2000), No. 1, 150-151.

[5] Jozsef Sandor, On certain inequalities involving the Smarandache dunction, Scientia Magna, 2(2006), No. 3, 78-80.

[6] J. Fu, An equaltion involving the Smarandache function, Scientia Magna, 2(2006), No. 4, 83-86.

[7] L. K. Hua, Itroduction to number theory, Springer-Verlag, 1982.