Large sieve inequality for sums of Legendre symbols over short intervals

Abstract

We use the Burgess bound and Selberg sieve to obtain an upper bound on the second moment of sums over an interval $[u+1,u+h]$ of Legendre symbols modulo primes $p$ in a dyadic interval $[Q,2Q]$. The bound is nontrivial and gives a power saving with respect to $h$ for any $u \le Q$, provided $h \ge \psi(Q)$ for any function $\psi(Q)\to\infty$ as $Q\to\infty$. This can be viewed as a generalisation of a result of D. R. Heath-Brown (1995) on moments of sums or quadratic characters over the initial interval $[1,h]$.