Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 909-916, 2011

On Lehmer's problem and Dedekind sums

Xiaowei Pan, Wenpeng Zhang

Xiaowei Pan, Wenpeng Zhang, Department of Mathematics, Northwest University, Xi'an, Shaanxi, 710069, P. R. China; Xi'an Medical University, Xi'an, Shaanxi, P. R. China, e-mail: wpzhang@nwu.edu.cn

Abstract: Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \leq p-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq p-1$ such that $ab \equiv c $ (mod $p$). Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $p$) for $1 \le a$, $b \leq p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv1\pmod p$. 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 $N(c,p)-\frac12\phi(p)$ and the Dedekind sums $S(c,p)$, and to establish a sharp asymptotic formula for it.

Keywords: Lehmer's problem, error term, Dedekind sums, hybrid mean value, asymptotic formula

Classification (MSC 2010): 11L40, 11F20


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