A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N . p . 1. Denote by r(N, c; p) the number of integers a satisfying 1 . a . N and 2 . a + ., where . is an integer with 1 . . . p . 1, a. . c (mod p). It is well known that r(N, c; p) = 1/2N + O(p 1/2log2 p). The main purpose of this paper is to give an asymptotic formula for . p.1 c=1 (r(N, c; p) . 1/2N)2.