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On the Smarandache ceil function and the Dirichlet divisor function.

[section] 1. Introduction

For a fixed positive integer k and any positive integer n, the Smarandache ceil function [S.sub.k] (n) is defined as

{[S.sub.k](n) = min m [member of] N : n|[m.sup.k]}.

This function was introduced by Professor Smarandache. About this function, many scholars studied its properties. Ibstedt [2] presented the following property: ([for all]a, b [member of] N)(a, b) = 1 [??] [S.sub.k](ab) = [S.sub.k](a)[S.sub.k](b). It is easy to see that if (a, b) = 1, then ([S.sub.k](a), [S.sub.k](b)) = 1.

In her thesis, Ren Dongmei [4] proved the asymptotic formular

[summation over (n [less than or equal to] x)] d([S.sub.k](n)) = [c.sub.1]x log x + [c.sub.2]x + O([x.sup.1/2+[epsilon]]), (1)

where [c.sub.1] and [c.sub.2] are computable constants, and [epsilon] is any fixed positive number.

The aim of this short note is to prove the following

Theorem. Let [d.sub.3] (n) denote the Piltz divisor function of dimensional 3, then for any real number x [greater than or equal to] 2, we have

[summation over (n [less than or equal to] x)] [d.sub.3]([S.sub.k](n)) = x[P.sub.2,k](log x) + O([x.sup.1/2][e.sup.-c[delta](x)]), (2)

where [P.sub.2,k](log x) is a polynomial of degree 2 in log x , [delta](x) := [log.sup.3/5] x[(log log x).sup.-1/5] , c > 0 is an absolute constant.

Remark. The estimate O([x.sup.1/2+[epsilon]]) in (1) can also be improved to O([x.sup.1/2][e.sup.-c[delta](x)]) by a similar approach.

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following two lemmas.

Lemma 1. Let f(n) be an arithmetical function for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.1] [greater than or equal to] [a.sub.2] [greater than or equal to] ... [greater than or equal to] [a.sub.l] > 1/c > a [greater than or equal to] 0, r [greater than or equal to] 0, [P.sub.1](t), ..., [P.sub.l] (t) are polynomials in t of degrees not exceeding r, and c [greater than or equal to] 1 and b [greater than or equal to] 1 are fixed integers. Suppose for Rs > 1 that

[[infinity].summation over (n = 1)] [[mu].sub.b](n)/[n.sup.s] = 1/[[zeta].sup.b](s).

If h(n) = [summation over ([d.sup.c]|n)] [[mu].sub.b](d)f(n/[d.sup.c]), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [R.sub.1](t) ... [R.sub.l](t) are polynomials in t of degrees not exceeding r, and for some D > 0

[E.sub.c](x) [much less than] [x.sup.1/c] exp(-D[(log x).sup.3/5][(log log x).sup.-1/5]). (4)

Proof. If b = 1, Lemma 1 is Theorem 14.2 of Ivic[3]. When b [greater than or equal to] 2, Lemma 1 can be proved by the same approach.

Lemma 2. Let f (m), g (n) be arithmetical functions such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[alpha].sub.1] [greater than or equal to] [[alpha].sub.2] [greater than or equal to] ... [greater than or equal to] [[alpha].sub.J] > [alpha] > [beta] > 0, where [P.sub.j] (t) is polynomial in t. If h(n) = [summation over (n = md)] f(m)g(d), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [Q.sub.j](t)(j = 1, ... , J) are polynomials in t.

Now we prove our theorem, which is closely related to the Piltz divisor problem. Let [[DELTA].sub.3](x) denotes the error term in the asymptotic formula for [summation over (n [less than or equal to] x)] [d.sub.3](n). We know that

[D.sub.3](x) = [summation over (n [less than or equal to x])] [d.sub.3](n) = x[H.sub.3](log x) + [[DELTA].sub.3](x), (6)

where [H.sub.3](u) is a polynomial of degree 2 in u. For the upper bound of [[DELTA].sub.3](x), Kolesnik[1] proved that

[[DELTA].sub.3](x) [much less than] [x.sup.43/96 + [epsilon]]. (7)

Let f(s) = [[infinity].summation over (n = 1)] [d.sub.3]([S.sub.k](n))/[n.sup.s] (Res > 1). By the Euler product formula we get for Rs > 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

It is easy to prove that [G.sub.k](s) is absolutely convergent for Res > 1/3.

Let [[zeta].sup.3](s)[G.sub.k](s) = [[infinity].summation over (n = 1)] [f.sub.k](n)/[n.sup.s] (Rs > 1). By Lemma 2 and (2.4) we can get

[summation over (n [less than or equal to] x)] [f.sub.k](n)= x[M.sub.3](log x) + O([x.sup.43/96 + [epsilon]]), (9)

where [M.sub.3](u) is a polynomial of degree 2 in u. Then we can get

[summation over (n [less than or equal to] x)] [absolute value of [f.sub.k](n)] [much less than] x [log.sup.2] x. (10)

We konw 1/[[zeta].sup.3](s) = [[infinity].summation over (n = 1)] [[mu].sub.3](d)/[d.sup.s] (Rs > 1). From (8) we have the relation

[d.sub.3]([S.sub.k](n)) = [summation over (n = m[d.sup.2])] [f.sub.k](m)[[mu].sub.3](d). (11)

Now Theorem follows from (9)-(11) with the help of Lemma 1.

References

[1] G. Kolesnik, On the estimation of multiple exponential sums, in Recent Progress in Analytic Number Theory, Symposium Durham, Academic, London, 1981a, 1(1979), 231-246.

[2] Ibstedt, Surfining On the Ocean of Number-A New Smarandache Notions and Similar Topics, New Mexico, Erhus University Press.

[3] A. Ivic, The Riemann Zeta-function, John Wiley & Sons, 1985.

[4] Ren Dongmei, Doctoral thesis, Xi'an Jiaotong University, 2006.

(1) This work is supported by National Natural Science Foundation of China(Grant No. 10771127) and Mathematical Tianyuan Foundation of China (Grant No. 10826028).

Lulu Zhang, Meimei Lu and Wenguang Zhai School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China lvmeimei2001@126.com, zhanglulu0916@163.com, zhaiwg@hotmail.com
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Title Annotation:Florentin Smarandache
Author:Zhang, Lulu; Lu, Meimei; Zhai, Wenguang
Publication:Scientia Magna
Article Type:Report
Geographic Code:9CHIN
Date:Dec 1, 2008
Words:1064
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