On Lehmer’s problem and Dedekind sums

被引:0
作者
Xiaowei Pan
Wenpeng Zhang
机构
[1] Northwest University,Department of Mathematics
[2] Xi’an Medical University,undefined
来源
Czechoslovak Mathematical Journal | 2011年 / 61卷
关键词
Lehmer’s problem; error term; Dedekind sums; hybrid mean value; asymptotic formula; 11L40; 11F20;
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学科分类号
摘要
Let p be an odd prime and c a fixed integer with (c, p) = 1. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one b with 0 ⩽ b ⩽ p − 1 such that ab ≡ c (mod p). Let N(c, p) denote the number of all solutions of the congruence equation ab ≡ c (mod p) for 1 ⩽ a, b ⩽ p−1 in which a and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline b $$\end{document} are of opposite parity, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline b $$\end{document} is defined by the congruence equation b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline b $$\end{document} ≡ 1 (mod p). The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L-functions to study the hybrid mean value problem involving N(c, p)−½φ(p) and the Dedekind sums S(c, p), and to establish a sharp asymptotic formula for it.
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页码:909 / 916
页数:7
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共 12 条
[1]  
Carlitz L.(1953)The reciprocity theorem for Dedekind sums Pac. J. Math. 3 523-527
[2]  
Conrey J.B.(1996)Mean values of Dedekind sums J. Number Theory 56 214-226
[3]  
Fransen E.(2001)On the mean value of Dedekind sums J. Number Theory 87 173-188
[4]  
Klein R.(2006)On a problem of D.H. Lehmer over short intervals J. Math. Anal. Appl. 320 756-770
[5]  
Scott C.(2000)A note on the mean square value of the Dedekind sums Acta Math. Hung. 86 275-289
[6]  
Jia C.H.(2003)A problem of D.H. Lehmer and its mean square value formula Jap. J. Math. 29 109-116
[7]  
Xu Z.F.(1993)On a problem of D.H. Lehmer and its generalization Compos. Math. 86 307-316
[8]  
Zhang W.P.(1996)On the mean values of Dedekind sums J. Théor. Nombres Bordx. 8 429-442
[9]  
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