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 withN ≤ p - 1. Denote by r（N, c;p） the number of integers a satisfying 1 ≤ a ≤ N and 2 a ＋ a, where a is an integer with 1 ≤a≤ p - 1, aa ≡ c （mod p）. It is well known that r（N, c;p） = 1/2N ＋ O（p1/2log2p）.The main purpose of this paper is to give an asymptotic formula for ∑p-1 c=1（τ（N,c;p）-1/2N）2.