Large sums of high order characters II

TL;DR

This paper establishes new upper bounds for the distribution of primitive characters of large order, improving previous results by removing restrictions on the modulus and order, and refines estimates for character sums.

Contribution

It introduces novel bounds for character level sets and enhances mean-squared estimates for character sums beyond Burgess' theorem for large order characters.

Findings

Non-trivial bounds for character level sets when $x > q^{ ext{delta}}$.

Refined mean-squared estimates for sums of $oldsymbol{ ext{chi}^ ext{l}}$ beyond Burgess.

Improved Pólya-Vinogradov inequality for characters with large even order.

Abstract

Let $\chi$ be a primitive character modulo $q$, and let $\delta > 0$. Assuming that $\chi$ has large order $d$, for any $d$th root of unity $\alpha$ we obtain non-trivial upper bounds for the number of $n \leq x$ such that $\chi(n) = \alpha$, provided $x > q^{\delta}$. This improves upon a previous result of the first author by removing restrictions on $q$ and $d$. As a corollary, we deduce that if the largest prime factor of $d$ satisfies $P^+(d) \to \infty$ then the level set $\chi(n) = \alpha$ has $o(x)$ such solutions whenever $x > q^{\delta}$, for any fixed $\delta > 0$. Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters $\chi^\ell$, averaged over $1 \leq \ell \leq d-1$, due to the first author, which goes beyond Burgess' theorem as soon as $d$ is sufficiently large. We in fact show the alternative result that either (a) the partial sum of $\chi$ itself, or (b) the partial sum of $\chi^\ell$, for ``almost all'' $1 \leq \ell \leq d-1$, exhibits cancellation on the interval $[1,q^{\delta}]$, for any fixed $\delta > 0$. By an analogous method, we also show that the P\'{o}lya-Vinogradov inequality may be improved for either $\chi$ itself or for almost all $\chi^\ell$, with $1 \leq \ell \leq d-1$. In particular, our averaged estimates are non-trivial whenever $\chi$ has sufficiently large even order $d$.