Quadratic Non-residues in Short Intervals

TL;DR

This paper establishes an upper bound on the number of primes within a dyadic interval that lack quadratic non-residues in a specified short interval, extending classical results on quadratic non-residues modulo primes.

Contribution

It introduces a novel bound using the Burgess bound and combinatorial sieve for primes with no quadratic non-residues in short intervals, generalizing previous average-case estimates.

Findings

Bound is nontrivial for any increasing function s Q pproaches infinity.

Extends classical estimates on the smallest quadratic non-residue to a broader setting.

Provides a new perspective on the distribution of quadratic non-residues among primes.

Abstract

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+\psi(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $\psi(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.