Journal of the
Korean Mathematical Society JKMS

Let $q\ge 3$ be an odd integer and $a$ be an integer coprime to $q$. Denote by $N(a,q)$ the number of pairs of integers $b,c$ with $bc\equiv a \pmod q$, $1\leq b,c\leq\frac{q-1}{2}$ and with $b,c$ having different parity. The main purpose of this paper is to study the sum $\mathop{{\sum}'}_{a=1}^q\left(N(a,q)-\frac{\phi(q)}{8}\right)^2$, and obtain a sharp asymptotic formula.

Keywords: error term, mean value, asymptotic formula

MSC numbers: 11B50