Quadratic non-residues in short intervals

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 non-trivial 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 $u=0$.