A new mean value related to D. H. Lehmer's problem and Kloosterman sums

摘要

Let $q>1$ be an odd integer and $c$ be a fixed integer with $(c, q)=1$. For each integer $a$ with $1le a leq q-1$, it is clear that there exists one and only one $b$ with $0leq b leq q-1$ 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 $boverline{b}equiv 1$ $(mod q)$. 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 $left(N(c, q)-rac{1}{2}phi(q)ight)$ and Kloosterman sums, and give a sharper asymptotic formula for it.