Dr Sacha Mangerel alexander.mangerel@durham.ac.uk
Assistant Professor
Let χ $\chi$ be a primitive character modulo a prime q $q$ , and let δ > 0 $\delta > 0$ . It has previously been observed that if χ $\chi$ has large order d ⩾ d 0 ( δ ) $d \geqslant d_0(\delta)$ then χ ( n ) ≠ 1 $\chi (n) \ne 1$ for some n ⩽ q δ $n \leqslant q^{\delta}$ , 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 $d$ is squarefree then for any d $d$ th root of unity α $\alpha$ the number of n ⩽ x $n \leqslant x$ such that χ ( n ) = α $\chi (n) = \alpha$ is o d → ∞ ( x ) $o_{d \rightarrow \infty}(x)$ whenever x > q δ $x > q^\delta$ . Consequently, when χ $\chi$ has sufficiently large order the sequence ( χ ( n ) ) n ⩽ q δ $(\chi (n))_{n \leqslant q^\delta}$ cannot cluster near 1 $\hskip.001pt 1$ for any δ > 0 $\delta > 0$ . Our proof relies on a second moment estimate for short sums of the characters χ ℓ $\chi ^\ell$ , averaged over 1 ⩽ ℓ ⩽ d − 1 $1 \leqslant \ell \leqslant d-1$ , that is non‐trivial whenever d $d$ has no small prime factors. In particular, given any δ > 0 $\delta > 0$ we show that for all but o ( d ) $o(d)$ powers 1 ⩽ ℓ ⩽ d − 1 $1 \leqslant \ell \leqslant d-1$ , the partial sums of χ ℓ $\chi ^\ell$ exhibit cancellation in intervals n ⩽ q δ $n \leqslant q^\delta$ as long as d ⩾ d 0 ( δ ) $d \geqslant d_0(\delta)$ 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 ⩽ d ⩽ q − 1 $3 \leqslant d \leqslant q-1$ , the Pólya–Vinogradov inequality may be improved for χ ℓ $\chi ^\ell$ on average over 1 ⩽ ℓ ⩽ d − 1 $1 \leqslant \ell \leqslant d-1$ , extending work of Granville and Soundararajan.
Mangerel, A. P. (2024). Large sums of high‐order characters. Journal of the London Mathematical Society, 109(1), Article e12841. https://doi.org/10.1112/jlms.12841
Journal Article Type | Article |
---|---|
Acceptance Date | Oct 27, 2023 |
Online Publication Date | Dec 19, 2023 |
Publication Date | Jan 1, 2024 |
Deposit Date | Nov 6, 2023 |
Publicly Available Date | Feb 28, 2024 |
Journal | Journal of the London Mathematical Society |
Print ISSN | 0024-6107 |
Electronic ISSN | 1469-7750 |
Publisher | Wiley |
Peer Reviewed | Peer Reviewed |
Volume | 109 |
Issue | 1 |
Article Number | e12841 |
DOI | https://doi.org/10.1112/jlms.12841 |
Public URL | https://durham-repository.worktribe.com/output/1819276 |
Publisher URL | https://londmathsoc.onlinelibrary.wiley.com/journal/14697750 |
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© 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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