Products of quadratic residues and related identities

Abstract

In this paper 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 $p\equiv 5(\mathrm{mod}8)$, then $$\prod _{0<x<p/2,(\frac{x}{p})=1}x\equiv (-1{)}^{1+r}(\mathrm{mod}p),$$ where $(\frac{\cdot }{p})$ 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 class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.