[{"abstract":[{"lang":"eng","text":"We analyze the average behavior of various arithmetic functions at the values of degree 𝑑 binary forms ordered by height, with probability 1. This approach yields averaged versions of the Chowla conjecture and the Bateman–Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Châtelet varieties defined by a fixed norm form of degree 𝑒 and by varying binary forms of fixed degree 𝑑, provided 𝑒 divides 𝑑. This proves an average version of a conjecture of Colliot-Thélène."}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1017/S0017089526101074"}],"year":"2026","PlanS_conform":"1","page":"1-34","corr_author":"1","mathsc":["11N32","11N37","11D57","11G35"],"status":"public","article_processing_charge":"Yes (via OA deal)","das_tickbox":"0","date_created":"2026-08-04T06:15:04Z","supplementarymaterial":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","date_published":"2026-07-21T00:00:00Z","ddc":["500"],"oa":1,"quality_controlled":"1","external_id":{"arxiv":["2506.18065"]},"publisher":"Cambridge University Press","arxiv":1,"date_updated":"2026-08-04T06:28:01Z","publication":"Glasgow Mathematical Journal","article_type":"original","month":"07","researchdata_availability":"no","has_accepted_license":"1","publication_identifier":{"issn":["0017-0895"],"eissn":["1469-509X"]},"day":"21","department":[{"_id":"TiBr"},{"_id":"GradSch"}],"_id":"22644","publication_status":"epub_ahead","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"oa_version":"Published Version","type":"journal_article","title":"Liouville function, von Mangoldt function, and norm forms at random binary forms","citation":{"ista":"Diao Y. 2026. Liouville function, von Mangoldt function, and norm forms at random binary forms. Glasgow Mathematical Journal., 1–34.","ieee":"Y. Diao, “Liouville function, von Mangoldt function, and norm forms at random binary forms,” <i>Glasgow Mathematical Journal</i>. Cambridge University Press, pp. 1–34, 2026.","mla":"Diao, Yijie. “Liouville Function, von Mangoldt Function, and Norm Forms at Random Binary Forms.” <i>Glasgow Mathematical Journal</i>, Cambridge University Press, 2026, pp. 1–34, doi:<a href=\"https://doi.org/10.1017/s0017089526101074\">10.1017/s0017089526101074</a>.","short":"Y. Diao, Glasgow Mathematical Journal (2026) 1–34.","apa":"Diao, Y. (2026). Liouville function, von Mangoldt function, and norm forms at random binary forms. <i>Glasgow Mathematical Journal</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0017089526101074\">https://doi.org/10.1017/s0017089526101074</a>","ama":"Diao Y. Liouville function, von Mangoldt function, and norm forms at random binary forms. <i>Glasgow Mathematical Journal</i>. 2026:1-34. doi:<a href=\"https://doi.org/10.1017/s0017089526101074\">10.1017/s0017089526101074</a>","chicago":"Diao, Yijie. “Liouville Function, von Mangoldt Function, and Norm Forms at Random Binary Forms.” <i>Glasgow Mathematical Journal</i>. Cambridge University Press, 2026. <a href=\"https://doi.org/10.1017/s0017089526101074\">https://doi.org/10.1017/s0017089526101074</a>."},"OA_place":"publisher","OA_type":"hybrid","author":[{"orcid":"0000-0002-4989-5330","id":"7b7eb4ca-eb2c-11ec-b98b-accec0b20c3b","full_name":"Diao, Yijie","first_name":"Yijie","last_name":"Diao"}],"acknowledgement":"I am deeply grateful to my advisor Tim Browning for suggesting this problem and for the many valuable discussions that shaped this work. I would also like to thank Efthymios Sofos, Matteo Verzobio, and Shuntaro Yamagishi for discussions and insights that contributed to this paper. I am also very grateful to the anonymous referee for their careful reading and for the considerable effort they put into improving the manuscript.","doi":"10.1017/s0017089526101074"}]