{"scopus_import":"1","publication":"Journal für die reine und angewandte Mathematik","acknowledgement":"Tim Browning was supported by an FWF grant (DOI 10.55776/P32428), Efthymios Sofos was supported by EPSRC New Horizons grant EP/V048236/1, and Joni Teräväinen was supported by Academy of Finland grant no. 340098, a von Neumann Fellowship (NSF grant DMS-1926686), and funding from European Union’s Horizon Europe research and innovation programme under Marie Skłodowska-Curie grant agreement no. 101058904. This material is based upon work supported by a grant from the Institute for Advanced Study School of Mathematics. The authors are extremely grateful to the anonymous referee and to Yijie Diao for useful comments.","publication_identifier":{"eissn":["1435-5345"],"issn":["0075-4102"]},"OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"author":[{"orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","first_name":"Timothy D"},{"full_name":"Sofos, Efthymios","last_name":"Sofos","first_name":"Efthymios"},{"full_name":"Teräväinen, Joni","first_name":"Joni","last_name":"Teräväinen"}],"status":"public","date_created":"2026-09-06T22:01:58Z","title":"Bateman-Horn, polynomial Chowla and the Hasse principle with probability","_id":"22821","abstract":[{"text":"With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree 𝑑 polynomials 𝑓∈ℤ⁢[𝑡] that are ordered by height. This allows us to establish averaged versions of the Bateman–Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of 𝑓, both with arbitrary logarithmic power savings.","lang":"eng"}],"oa_version":"Preprint","type":"journal_article","article_type":"original","citation":{"short":"T.D. Browning, E. Sofos, J. Teräväinen, Journal Für Die Reine Und Angewandte Mathematik (2026).","ista":"Browning TD, Sofos E, Teräväinen J. 2026. Bateman-Horn, polynomial Chowla and the Hasse principle with probability. Journal für die reine und angewandte Mathematik.","ieee":"T. D. Browning, E. Sofos, and J. Teräväinen, “Bateman-Horn, polynomial Chowla and the Hasse principle with probability,” <i>Journal für die reine und angewandte Mathematik</i>. De Gruyter, 2026.","ama":"Browning TD, Sofos E, Teräväinen J. Bateman-Horn, polynomial Chowla and the Hasse principle with probability. <i>Journal für die reine und angewandte Mathematik</i>. 2026. doi:<a href=\"https://doi.org/10.1515/crelle-2026-0062\">10.1515/crelle-2026-0062</a>","mla":"Browning, Timothy D., et al. “Bateman-Horn, Polynomial Chowla and the Hasse Principle with Probability.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>, De Gruyter, 2026, doi:<a href=\"https://doi.org/10.1515/crelle-2026-0062\">10.1515/crelle-2026-0062</a>.","apa":"Browning, T. D., Sofos, E., &#38; Teräväinen, J. (2026). Bateman-Horn, polynomial Chowla and the Hasse principle with probability. <i>Journal Für Die Reine Und Angewandte Mathematik</i>. De Gruyter. <a href=\"https://doi.org/10.1515/crelle-2026-0062\">https://doi.org/10.1515/crelle-2026-0062</a>","chicago":"Browning, Timothy D, Efthymios Sofos, and Joni Teräväinen. “Bateman-Horn, Polynomial Chowla and the Hasse Principle with Probability.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>. De Gruyter, 2026. <a href=\"https://doi.org/10.1515/crelle-2026-0062\">https://doi.org/10.1515/crelle-2026-0062</a>."},"project":[{"grant_number":"P32428","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"New frontiers of the Manin conjecture"}],"date_updated":"2026-09-09T13:28:45Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","corr_author":"1","oa":1,"day":"25","publisher":"De Gruyter","month":"08","department":[{"_id":"TiBr"}],"external_id":{"arxiv":["2212.10373"]},"doi":"10.1515/crelle-2026-0062","publication_status":"epub_ahead","quality_controlled":"1","year":"2026","article_processing_charge":"No","date_published":"2026-08-25T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.10373"}],"arxiv":1,"fulldoi":"https://doi.org/10.1515/crelle-2026-0062"}