On a problem of D.H. Lehmer over short intervals

Let p be an odd prime and a be an integer coprime to p. Denote by N(a,p) the number of pairs of integers b, c with bc ≡ a (mod p), 1 b, c (p −1)/2 and with b, c having different parity. The main purpose of this paper is to study the sum p−1 a=1 (N(a,p) − (p −1)/8)2, and obtain a sharp asymptotic formula. © 2005 Elsevier Inc. All rights reserved