Average sizes of mixed character sums
Wang V, Xu M. 2026. Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics., 1–15.
Download (ext.)
Journal Article
| Epub ahead of print
| English
Author
Wang, VictorISTA 
;
Xu, Max
;
Xu, MaxCorresponding author has ISTA affiliation
Department
Abstract
We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper.
Publishing Year
Date Published
2026-01-01
Journal Title
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Publisher
Cambridge University Press
Acknowledgement
We thank Ofir Gorodetsky, Andrew Granville, Adam Harper, Youness Lamzouri,
Kannan Soundararajan, Ping Xi, and Matt Young for their interest, helpful discussions, and comments. Special thanks are due to Jonathan Bober, Oleksiy Klurman,
and Besfort Shala for sending us a letter about Question 1.3, and to Hung Bui
for informing us of [7]. V.W. thanks Stanford University for its hospitality and is supported by the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska–Curie Grant Agreement No. 101034413. M.X. is supported by a Simons Junior Fellowship from the Simons Society of Fellows at the
Simons Foundation.
Page
1-15
ISSN
eISSN
IST-REx-ID
Cite this
Wang V, Xu M. Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2026:1-15. doi:10.1017/prm.2026.10123
Wang, V., & Xu, M. (2026). Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University Press. https://doi.org/10.1017/prm.2026.10123
Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University Press, 2026. https://doi.org/10.1017/prm.2026.10123.
V. Wang and M. Xu, “Average sizes of mixed character sums,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University Press, pp. 1–15, 2026.
Wang V, Xu M. 2026. Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics., 1–15.
Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Cambridge University Press, 2026, pp. 1–15, doi:10.1017/prm.2026.10123.
All files available under the following license(s):
Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
Link(s) to Main File(s)
Access Level
Open Access
Export
Marked PublicationsOpen Data ISTA Research Explorer
Sources
arXiv 2411.14181
Google Scholar