On Lehmer's problem and Dedekind sums

Pan, Xiaowei ; Zhang, Wenpeng

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 909-916.

Résumé

Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ 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 $1\le a$, $b\le p-1$ in which $a$ and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b\bar{b}\equiv 1(\mathrm{mod}p)$. 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 $N(c,p)-\frac{1}{2}\varphi (p)$ and the Dedekind sums $S(c,p)$, and to establish a sharp asymptotic formula for it.

Détail

DOI : 10.1007/s10587-011-0058-2

Classification : 11F20, 11L40, 11M20
Mots-clés : Lehmer's problem; error term; Dedekind sums; hybrid mean value; asymptotic formula