Large sums of high‐order characters

Mangerel, Alexander P.

Abstract

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.

Citation

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