Large sums of high order characters II

Abstract

Let $\ddot{I}‡$ be a primitive character modulo $q$, and let $\hat{I}´>0$. Assuming that $\ddot{I}‡$ has large order $d$, for any $d$th root of unity $\hat{I}\pm$ we obtain non-trivial upper bounds for the number of $n\le x$ such that $\ddot{I}‡(n)=\hat{I}\pm$, provided $x>q^{\hat{I}}´$. 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 $\ddot{I}‡(n)=\hat{I}\pm$ has $o(x)$ such solutions whenever $x>q^{\hat{I}}´$, for any fixed $\hat{I}´>0$. Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters $\ddot{I}{‡}^{\ell }$, averaged over $1\le \ell \le d-1$, 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 $\ddot{I}‡$ itself, or (b) the partial sum of $\ddot{I}{‡}^{\ell }$, for ``almost all'' $1\le \ell \le d-1$, exhibits cancellation on the interval $[1,q^{\hat{I}}´]$, for any fixed $\hat{I}´>0$. By an analogous method, we also show that the Pólya-Vinogradov inequality may be improved for either $\ddot{I}‡$ itself or for almost all $\ddot{I}{‡}^{\ell }$, with $1\le \ell \le d-1$. In particular, our averaged estimates are non-trivial whenever $\ddot{I}‡$ has sufficiently large even order $d$.

32 pages, comments welcome