Large sums of high‐order characters

Abstract

AbstractLet be a primitive character modulo a prime , and let . It has previously been observed that if has large order then for some , in analogy with Vinogradov's conjecture on quadratic non‐residues. We give a new and simple proof of this fact. We show, furthermore, that if is squarefree then for any th root of unity the number of such that is whenever . Consequently, when has sufficiently large order the sequence cannot cluster near for any . Our proof relies on a second moment estimate for short sums of the characters , averaged over , that is non‐trivial whenever has no small prime factors. In particular, given any we show that for all but powers , the partial sums of exhibit cancellation in intervals as long as is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime , the Pólya–Vinogradov inequality may be improved for on average over , extending work of Granville and Soundararajan.