Harper’s beyond square-root conjecture
Wang V, Xu MW. 2025. Harper’s beyond square-root conjecture. International Mathematics Research Notices. 2025(18), rnaf279.
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Author
Wang, VictorISTA 
;
Xu, Max Wenqiang
;
Xu, Max WenqiangDepartment
Abstract
We explain how the (shifted) Ratios Conjecture for $L(s,\chi )$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of “beyond square-root cancellation” for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.
Publishing Year
Date Published
2025-09-01
Journal Title
International Mathematics Research Notices
Publisher
Oxford University Press
Acknowledgement
The first author is supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101034413. The second author is supported by a Simons Junior Fellowship from Simons Foundation. We thank Paul Bourgade and Kannan Soundararajan for discussions on random matrices and probability, Alexandra Florea for helpful comments on the Ratios Conjecture, and Joni Teräväinen for providing several references. We are also grateful to Alexandra Florea, Adam Harper, Joni Teräväinen, and the referee for helpful comments on earlier drafts.
Volume
2025
Issue
18
Article Number
rnaf279
ISSN
eISSN
IST-REx-ID
Cite this
Wang V, Xu MW. Harper’s beyond square-root conjecture. International Mathematics Research Notices. 2025;2025(18). doi:10.1093/imrn/rnaf279
Wang, V., & Xu, M. W. (2025). Harper’s beyond square-root conjecture. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnaf279
Wang, Victor, and Max Wenqiang Xu. “Harper’s beyond Square-Root Conjecture.” International Mathematics Research Notices. Oxford University Press, 2025. https://doi.org/10.1093/imrn/rnaf279.
V. Wang and M. W. Xu, “Harper’s beyond square-root conjecture,” International Mathematics Research Notices, vol. 2025, no. 18. Oxford University Press, 2025.
Wang V, Xu MW. 2025. Harper’s beyond square-root conjecture. International Mathematics Research Notices. 2025(18), rnaf279.
Wang, Victor, and Max Wenqiang Xu. “Harper’s beyond Square-Root Conjecture.” International Mathematics Research Notices, vol. 2025, no. 18, rnaf279, Oxford University Press, 2025, doi:10.1093/imrn/rnaf279.
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