Products of quadratic residues and related identities

Abstract

. We study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let p be an odd prime. We prove that if p ≡ 5 (mod 8) , then where (cid:0) · p (cid:1) is the Legendre symbol and r is the number of 4 th power residues modulo p in the interval (0 , p/ 2) . Our work involves the class number formula, quartic Gauss sums, Stickelberger’s congruence and values of Dirichlet L-series at negative integers.