Mean Value Estimates of the Error Terms of Lehmer Problem

摘要

Let e?‘? be an odd prime and e?‘? be an integer coprime with e?‘?. Denote by $N(a,p)$ the number of pairs of integers e?‘?, e?‘? with $bca‰? a(mathrm{mod} p),1a‰¤ b, c p$ and with e?‘?, e?‘? having different parity. The main purpose of this paper is to study the mean square value problem of $N(a,p)-frac{1}{2}(p-1)$ over interval (e?‘?, e?‘?+e?‘€] with e?‘€,e?‘? positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.