Large Sums of High Order Characters

Abstract

Let $χ$ be a primitive character modulo a prime $q$, and let $δ> 0$. It has previously been observed that if $χ$ has large order $d \geq d_0(δ)$ then $χ(n) \neq 1$ for some $n \leq q^δ$, 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 $d$ is squarefree then for any $d$th root of unity $α$ the number of $n \leq x$ such that $χ(n) = α$ is $o_{d \to \infty}(x)$ whenever $x > q^δ$. Consequently, when $χ$ has sufficiently large order the sequence $(χ(n))_{n \leq q^δ}$ cannot cluster near $1$ for any $δ> 0$. Our proof relies on a second moment estimate for short sums of the characters $χ^\ell$, averaged over $1 \leq \ell \leq d-1$, that is non-trivial whenever $d$ has no small prime factors. In particular, given any $δ> 0$ we show that for all but $o(d)$ powers $1 \leq \ell \leq d-1$, the partial sums of $χ^\ell$ exhibit cancellation in intervals $n \leq q^δ$ as long as $d \geq d_0(δ)$ 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 $3 \leq d \leq q-1$, the Pólya-Vinogradov inequality may be improved for $χ^\ell$ on average over $1 \leq \ell \leq d-1$, extending work of Granville and Soundararajan.