Harper’s Beyond Square-Root Conjecture
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.
Authors
- Victor Y. Wang (ORCID: https://orcid.org/0000-0002-0704-7026)
- Max Wenqiang Xu (ORCID: https://orcid.org/0000-0002-4964-0186)
Institutions
- Institute of Science and Technology Austria (AT)
- Courant Institute of Mathematical Sciences (US)
- Institute of Mathematics, Academia Sinica (TW)
- Institute of Sociology, Academia Sinica (TW)
- Institute of Chemistry, Academia Sinica (TW)
Publication Details
- Journal
- International Mathematics Research Notices
- Published
- 2025-09-01
- DOI
- https://doi.org/10.1093/imrn/rnaf279
- Primary Topic
- Analytic Number Theory Research
- Type
- article
- Field-Weighted Citation Impact
- 0.00