Gaussian phenomena for small quadratic residues and non-residues

Abstract

Assuming the Generalized Riemann Hypothesis, it is known that the smallest quadratic non-residue modulo a prime p p is less than or equal to ( log ⁡ p ) 2 (\log p)^2 . Our aim in this paper is to establish the distribution of quadratic non-residues in even smaller intervals of size ( log ⁡ p ) A (\log p)^A with A > 1 A >1 , for almost all primes p p .