Di Han and Wenpeng Zhang Northwest University, Northwest University

Abstract : Let $q>1$ be an odd integer and $c$ be a fixed integer with $(c, q)=1$. For each integer $a$ with $1\le a \leq q-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq q-1$ such that $ab \equiv c $ (mod $q$). Let $N(c, q)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $q$) for $1 \le a, b \leq q-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1$ $(\bmod q)$. The main purpose of this paper is using the mean value theorem of Dirichlet $L$-functions to study the mean value properties of a summation involving $\left(N(c, q)-\frac{1}{2}\phi(q)\right)$ and Kloosterman sums, and give a sharper asymptotic formula for it.

Keywords : D. H. Lehmer's problem, error term, Kloosterman sums, hybrid mean value, asymptotic formula