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GENERALIZED D. H. LEHMER PROBLEM OVER SHORT INTERVALS

Published online by Cambridge University Press:  13 December 2010

PING XI  and
YUAN YI
PING XI
Affiliation:
School of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China email: xprime@163.com
YUAN YI
Affiliation:
School of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China and Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA email: yuanyi@mail.xjtu.edu.cn
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Abstract

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Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: \begin{displaymath} N(\mathcal{A},\mathcal{B},c,n,\ell;q)=\#\{(a,b)\in\mathcal{A}\times\mathcal{B}|ab\equiv c(\bmod \, q),(a+b,n)=\ell\}. \end{displaymath} The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

Keywords

Primary 11A0711N37Secondary 11L05

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 293 - 299
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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