A MEAN VALUE FUNCTION AND ITS COMPUTATIONAL FORMULA RELATED TO D. H. LEHMER'S PROBLEM

Abstract

Let p be an odd prime and c be a fixed integer with (c, p) = 1. For each integer a with <TEX>$1{\leq}a{\leq}p-1$</TEX>, it is clear that there exists one and only one b with <TEX>$0{\leq}b{\leq}p-1$</TEX> such that <TEX>$ab{\equiv}c$</TEX> mod p. Let N(c, p) denote the number of all solutions of the congruence equation <TEX>$ab{\equiv}c$</TEX> mod p for <TEX>$1{\leq}a$</TEX>, <TEX>$b{{\leq}}p-1$</TEX> in which a and <TEX>$\bar{b}$</TEX> are of opposite parity, where <TEX>$\bar{b}$</TEX> is defined by the congruence equation <TEX>$b{\bar{b}}{\equiv}1$</TEX> mod p. The main purpose of this paper is using the mean value theorem of Dirichlet L-functions and the properties of Gauss sums to study the computational problem of one kind mean value function related to <TEX>$E(c,p)=N(c,p)-{\frac{1}{2}}{\phi}(p)$</TEX>, and give its an exact computational formula.