Large sieve inequality for sums of Legendre symbols over short intervals

TL;DR

This paper establishes a new upper bound on the second moment of sums of Legendre symbols over short intervals, extending previous results and providing power savings under certain conditions.

Contribution

It introduces a novel bound using the Burgess bound and Selberg sieve, generalizing Heath-Brown's earlier work on quadratic character sums.

Findings

Bound is nontrivial and offers power savings for h ≥ ψ(Q)

Generalizes Heath-Brown's results to arbitrary starting points u

Applicable to sums over short intervals in prime moduli

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 $[1,h]$.