Poisson Behavior for Chains of Small Quadratic Non-Residues

Abstract

Abstract It is known that under the Generalized Riemann Hypothesis, the smallest quadratic non-residue modulo a prime $p$ is less than or equal to $(\log p)^{2}$. In ranges slightly larger, of size $(\log p)^{A}$ with $A>2$, we consider chains of $r$ consecutive quadratic non-residues. We prove unconditionally that for almost all primes $p$ in short intervals, these chains exhibit Poisson behavior.