# 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  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  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 .

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 .

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

 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.

 Ibstedt, Surfining on the ocean of number-A new Smarandache notions and similar topics, New Mexico, Erhus University Press.

 A. Ivic, The Riemann Zeta-function, John Wiley Sons, 1985.

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

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

 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).