Some new sums related to D. H. Lehmer problem

Czechoslovak Mathematical Journal

Han Zhang¹ 

[1] Wenpeng Zhang, School of Mathematics, Northwest University, Xuefu Avenue No. 1, Chang'an, Xi'an, Shaanxi, 710127, P. R. China

关键词: Lehmer number;    analytic method;    trigonometric sums;    asymptotic formula;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

About Lehmer's number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N(k; p)$ denote the number of all $1 \leq a_i \leq p - 1 $ such that $a_1a_2 \cdots a_k \equiv1 \mod p$ and $2 \mid a_i + \bar{a}_i + 1,$ $i = 1, 2, \cdots, k$. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function $N(k; p),$ and give an interesting asymptotic formula for it.

【 授权许可】

Unknown