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A short interval result for 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 cell 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 formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [c.sub.1] and [c.sub.2] are computable constants, and [member of] is any fixed positive number. The aim of this paper is to prove the following:

Theorem 1.1. Let d(n) denote the Dirichlet divisor function, [S.sub.k] (n) denote the Smarandache cell function, then for 1/4 < [theta] < 1/3, [x.sup.[theta]+2[member of] [less than or equal to] y [less than or equal to] x, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where H(x) = [t.sub.1]x log x + [t.sub.2]x.

Notations 1.1. Throughout this paper, e always denotes a fixed but sufficiently small positive constant.

[section]2. Proof of the Theorem

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

Lemma 2.1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The asymptotic formula (3) is the well-known Dirichlet divisor problem. The latest value of [theta] is [theta] = 131/416 proved by Huxley [6].

Lemma 2.2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. It follows from [absolute value of g(n)] [much less than] [n..sup.-[alpha]+[member of].

Lemma 2.3. Let k [greater than or equal to] 2 be a fixed integer, 1 < y [less than or equal to] x be large real numbers and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.

Then we have

B(x, y; k, [member of]) [much less than] [yx.sup.[-member of] + [x.sup.1/4].

Proof. This is Lemma 2.3 of Zhai [5].

Now we prove our theorem, which is closely related to the Dirichlet divisor problem.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] here d ([S.sub.k](n) is multiplicative and by Euler product formula we have for [sigma] > 1 that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we get G(s) = [[SIGMA].sup.[infinity].sub.n=1] g(n)/[n.sup.s] and by the properties of Dirichlet series, it is absolutely convergent for Rs > 1/3.

By the convolution method, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where d(n) is the divisor function. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of Lemma 2.1, the inner sum in [[SIGMA].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Inserting the above expression into [[SIGMA].sub.1] and after some easy calculations, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

For [[SIGMA].sub.2], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if we notice that [n.sub.2] > [x.sup.[member of], and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by lemma 2.3 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Now our theorem follows from (4) and (7).

References

[1] G. Kolesnik, On the estimation of multiple exponential sums, in Recent Progress in Analytic Number Theory, Symposium Durham, Academic, London, 1981, 1(1979), 231-248.

[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.

[5] W. G. Zhai, Square-free numbers as sums of two squares, in Number Theory, Developments in Mathematics, Springer, 15(2006), 219-227.

[6] M. N. Huxley, Exponential sums and Lattice points III, Proc. London Math. Soc., 87(2003), No. 3, 591-609.

YingYing Zhang ([dagger]), Huafeng Liu ([double dagger]) and Peimin Zhao (#)

([dagger][double dagger]) School of mathematical Sciences, Shandong Normal University, Jinan, 250014, Shandong, P. R. China

(#) Linyi University, Feixian, 273400, Shandong, P. R. China

E-mail: zhangyingyingml@126.com liuhuafeng 19618@126.com zpm67101028@sina.com

(1) This work is supported by Natural Science Foundation of China (Grant No: 11001154), and Natural Science Foundation of Shandong Province (Nos: BS2009SF018, ZR2010AQ009).
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Author:Zhang, Yingying; Liu, Huafeng; Zhao, Peimin
Publication:Scientia Magna
Date:Sep 1, 2012
Words:773
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