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arXiv · 2405.04094

Harper's beyond square-root conjecture

Victor Y. Wang·Max Wenqiang Xu
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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.

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Victor Y. Wang, Max Wenqiang Xu. 2024-05-07. Harper's beyond square-root conjecture. https://doi.org/10.1093/imrn%2Frnaf279

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