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AN ARITHMETICAL FUNCTION AND ITS MEAN VALUE

Ding Liping Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China Abstract. A new arithmetical function is introduced, and an interesting asymptotic formula on its mean value is given.

1. Introduction For any positive integer n and any fixed positive integer k, we define the arithmetical function S k (n) as follows: S k (n) = max{x ∈ N | xk | n}. Because (∀a, b ∈ N ∗ )(a, b) = 1 ⇒ S k (ab) = max{x ∈ N | xk | a} · max{x ∈ N | xk | b} = S k (a) · S k (b), and

α

S k (pα ) = pb k c , where bxc denotes the greatest integer less than or equal to x. Therefore, if n = αr 1 α2 pα 1 p2 . · · · .pr is the prime powers decomposition of n, then we have αr b 1 α2 S k (pα 1 p2 . · · · .pr ) = p

α1 k

c

. · · · .pb

αr k

c

αr 1 = S ( pα 1 ). · · · .S ( pr ).

So S k is a multiplicative function. There are close relations between this function and the Smarandache ceil function [1]. In this paper, we study the mean value properties of S k , and give a sharp asymptotic formula for it. That is, we shall prove the following: Theorem 1. Let x ≥ 2. Then for any fixed positive integer k > 2, we have the asymptotic formula   1  Y X 1 x ζ(k − 1) 1− k S k (n) = + O x 2 + , ζ(2k − 2) p +p p n≤x

where ζ(s) is the Riemann zeta-function,

Y

denotes the product over all prime p,

p

and  be any fixed positive integer.

Key words and phrases. Arithmetical function; Mean value; Asymptotic formula.. *This work is supported by the N.S.F.(10271093) and P.N.S.F of P.R.China Typeset by AMS-TEX

1

2

DING LIPING

Theorem 2. For any real number x ≥ 2, we have the asymptotic formula    1  X 1 3x ln x Y 1 − S 2 (n) = + Cx + O x 2 + , 2+p π2 p p n≤x

where C is a computable constant. 2. Some Lemmas To complete the proof of the theorems, we need the following famous Perron formula [2]: Lemma. Suppose that the Dirichlet series f (s) =

∞ X

a(n)n−s , s = σ+it, converge

n=1

absolutely for σ > β, and that there exist a positive λ and a positive increasing function A(s) such that ∞ X

|a(n)| n−σ  (σ − β)−1 , σ → β + 0

n=1

and a(n)  A(n), n = 1, 2, · · · . Then for any b > 0, b + σ > β, and x not to be an integer, we have     Z b+iT X 1 xω A(2x)x1−σ log x xb −s0 = a(n)n f (s0 +ω) dω+O +O , 2πi b−iT ω T (b + σ − β)λ T || x ||

n≤x

where || x || is the nearest integer to x. 3. Proof of the Theorems In this section, we complete the proof of the Theorems. First we prove Theorem 1. Let ∞ X S k (n) f (s) = , ns n=1

Re(s) > 1. Then by the Euler product formula [3] and the multiplicative property of S k (n) we have f (s) =

Y p

S k (p) S k (p2 ) S k (pk ) 1+ + + · · · + +··· ps p2s pks



1 p p p2 1 1 p = 1 + s + 2s + · · · + (k−1)s + ks + (k+1)s + · · · + (2k−1)s + 2ks + · · · p p p p p p p p ! 1 1 1 Y p 1 − pks p2 1 − pks 1 1 − p(k−1)s + ks + 2ks ··· = 1+ s p p 1 − p1s p 1 − p1s 1 − p1s p ! 1 1 Y 1 − p1ks pks−1 1 1 − p(k−1)s + = 1+ s 1 p 1 − 1 − p1s 1 − p1ks s p p Y



AN ARITHMETICAL FUNCTION AND ITS MEAN VALUE

3

 1 = ζ(s) 1 + ks−1 − ks p p p  Y 1 ζ(s) ζ(ks − 1) 1 − ks = ζ(2ks − 2) p +p p Y

1

where ζ(s) is the Riemann zeta-function. So by Perron formula, with s0 = 0, b = 23 , T = x, we have X

n≤x

1 S k (n) = 2iπ

Z

3 2 +iT 3 2 −iT

where R(s) =

1 ζ(s)ζ(ks − 1) xs R(s) ds + O(x 2 +ε ), ζ(2ks − 2) s

Y p

1 1 − ks p +p



.

To estimate the main term 1 2iπ

Z

3 2 +iT 3 2 −iT

ζ(s)ζ(ks − 1) xs R(s) ds ζ(2ks − 2) s

we move the integral line from s = 32 ± iT to s = 12 ± iT . This time, when k > 2 the function ζ(s)ζ(ks − 1)xs R(s) f (s) = ζ(2ks − 2)s x ζ(2k−2) ζ(k

has a simple pole point at s = 1 with residue 1 2iπ

Z

3 2 +iT 3 2 −iT

+

Z

1 2 +iT

+

3 2 +iT

x = ζ(2k − 2) ζ(k − 1)

Z

Y p

1 2 −iT

+

1 2 +iT

Z

3 2 −iT 1 2 −iT

1 1 − ks p +p

!

− 1)R(1). So we have

ζ(s)ζ(ks − 1)xs R(s)ds ζ(2ks − 2)s



Note that 1 2iπ

Z

1 2 +iT 3 2 +iT

+

Z

1 2 −iT 1 2 +iT

+

Z

3 2 −iT 1 2 −iT

!

1 ζ(s)ζ(ks − 1)xs R(s)ds  x 2 + ζ(2ks − 2)s

From above we immediately get the asymptotic formula:   1  X Y x 1 S k (n) = + O x 2 + . ζ(k − 1) 1− k ζ(2k − 2) p +p p n≤x

This completes the proof of Theorem 1. If k = 2, then the function ζ(s)ζ(ks − 1)xs R(s) f (s) = ζ(2ks − 2)s

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DING LIPING

has a second order pole point at s = 1 with residue 0 xs lim [(s − 1) f (s)] ≡ lim (s − 1) g(s) s→1 s→1 s    0 xs sxs ln x − xs 2 2 = lim (s − 1) g(s) + (s − 1) g(s) . s→1 s s2 2

0

Note that ζ(2) =

π2 6



2

and lim (s − 1)2 g(s) =

s→1

3 , π2

from the above we have X

n≤x

   1  1 3x ln x Y 1 − S 2 (n) = + Cx + O x 2 + , 2+p π2 p p

where C is a computable constant. This completes the proof of Theorem 2. Acknowledgments The author express her gratitude to her supervisor Professor Zhang Wenpeng for his very helpful and detailed instructions. References 1. S. Tabirca and T. Tabirca, Smarandache notions journal 13 (2002), 30-36. 2. Pan Chengdong and Pan Chengbiao, Goldbach conjecture, Science Press, Beijing, 1992, pp. 145. 3. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.