{"type":"journal_article","doi":"10.1093/imrn/rnaf279","publication":"International Mathematics Research Notices","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"      2025","day":"01","author":[{"full_name":"Wang, Victor","first_name":"Victor","last_name":"Wang","orcid":"0000-0002-0704-7026","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9"},{"full_name":"Xu, Max Wenqiang","last_name":"Xu","first_name":"Max Wenqiang"}],"article_type":"original","publication_status":"published","publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"scopus_import":"1","date_created":"2026-02-17T07:45:45Z","date_updated":"2026-02-18T07:41:56Z","department":[{"_id":"TiBr"}],"volume":2025,"ec_funded":1,"external_id":{"arxiv":["2405.04094"]},"publisher":"Oxford University Press","oa_version":"Preprint","year":"2025","_id":"21265","arxiv":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2405.04094"}],"project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"article_number":"rnaf279","issue":"18","citation":{"ista":"Wang V, Xu MW. 2025. Harper’s beyond square-root conjecture. International Mathematics Research Notices. 2025(18), rnaf279.","ieee":"V. Wang and M. W. Xu, “Harper’s beyond square-root conjecture,” <i>International Mathematics Research Notices</i>, vol. 2025, no. 18. Oxford University Press, 2025.","apa":"Wang, V., &#38; Xu, M. W. (2025). Harper’s beyond square-root conjecture. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnaf279\">https://doi.org/10.1093/imrn/rnaf279</a>","chicago":"Wang, Victor, and Max Wenqiang Xu. “Harper’s beyond Square-Root Conjecture.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2025. <a href=\"https://doi.org/10.1093/imrn/rnaf279\">https://doi.org/10.1093/imrn/rnaf279</a>.","short":"V. Wang, M.W. Xu, International Mathematics Research Notices 2025 (2025).","ama":"Wang V, Xu MW. Harper’s beyond square-root conjecture. <i>International Mathematics Research Notices</i>. 2025;2025(18). doi:<a href=\"https://doi.org/10.1093/imrn/rnaf279\">10.1093/imrn/rnaf279</a>","mla":"Wang, Victor, and Max Wenqiang Xu. “Harper’s beyond Square-Root Conjecture.” <i>International Mathematics Research Notices</i>, vol. 2025, no. 18, rnaf279, Oxford University Press, 2025, doi:<a href=\"https://doi.org/10.1093/imrn/rnaf279\">10.1093/imrn/rnaf279</a>."},"OA_place":"repository","abstract":[{"lang":"eng","text":"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."}],"OA_type":"green","quality_controlled":"1","status":"public","date_published":"2025-09-01T00:00:00Z","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.","oa":1,"language":[{"iso":"eng"}],"title":"Harper’s beyond square-root conjecture","month":"09"}