Short Character Sums and the Pólya–Vinogradov Inequality

Mangerel, Alexander P

Abstract

We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to max1≤t≤q∣∣∣∣∑n≤tχ(n)∣∣∣∣=oq→∞(q√logq) then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever t>qε⁠, i.e., ∑n≤tχ(n)=oq→∞(t) for all t>qε.
We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.

Citation

Mangerel, A. P. (2022). Short Character Sums and the Pólya–Vinogradov Inequality. The Quarterly Journal of Mathematics, 71(4), 1281–1308. https://doi.org/10.1093/qmath/haaa031

• Journal Article Type: Article
• Acceptance Date: Aug 6, 2020
• Online Publication Date: Dec 11, 2022
• Publication Date: Dec 11, 2022
• Deposit Date: Oct 20, 2021
• Journal: The Quarterly Journal of Mathematics
• Print ISSN: 0033-5606
• Electronic ISSN: 1464-3847
• Publisher: Oxford University Press
• Volume: 71
• Issue: 4
• Pages: 1281–1308
• DOI: https://doi.org/10.1093/qmath/haaa031
• Public URL: https://durham-repository.worktribe.com/output/1228977