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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.00544 |
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Table of Contents:
- Let $χ$ be a primitive character modulo $q$, and let $δ> 0$. Assuming that $χ$ has large order $d$, for any $d$th root of unity $α$ we obtain non-trivial upper bounds for the number of $n \leq x$ such that $χ(n) = α$, provided $x > q^δ$. 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 $χ(n) = α$ has $o(x)$ such solutions whenever $x > q^δ$, for any fixed $δ> 0$. Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters $χ^\ell$, 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 $χ$ itself, or (b) the partial sum of $χ^\ell$, for ``almost all'' $1 \leq \ell \leq d-1$, exhibits cancellation on the interval $[1,q^δ]$, for any fixed $δ> 0$. By an analogous method, we also show that the Pólya-Vinogradov inequality may be improved for either $χ$ itself or for almost all $χ^\ell$, with $1 \leq \ell \leq d-1$. In particular, our averaged estimates are non-trivial whenever $χ$ has sufficiently large even order $d$.