Large sums of high order characters II
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917655117561856 |
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| author | Mangerel, Alexander P. You, Yichen |
| author_facet | Mangerel, Alexander P. You, Yichen |
| 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 $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00544 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large sums of high order characters II Mangerel, Alexander P. You, Yichen Number Theory 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 $d$. |
| title | Large sums of high order characters II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.00544 |