[{"fulldoi":"https://doi.org/10.1017/prm.2026.10123","acknowledgement":"We thank Ofir Gorodetsky, Andrew Granville, Adam Harper, Youness Lamzouri,\r\nKannan Soundararajan, Ping Xi, and Matt Young for their interest, helpful discussions, and comments. Special thanks are due to Jonathan Bober, Oleksiy Klurman,\r\nand Besfort Shala for sending us a letter about Question 1.3, and to Hung Bui\r\nfor informing us of [7]. V.W. thanks Stanford University for its hospitality and is supported by the European Union’s Horizon 2020 research and innovation program\r\nunder the Marie Skłodowska–Curie Grant Agreement No. 101034413. M.X. is supported by a Simons Junior Fellowship from the Simons Society of Fellows at the\r\nSimons Foundation.","das_tickbox":"0","article_type":"original","publication_identifier":{"eissn":["1473-7124"],"issn":["0308-2105"]},"abstract":[{"lang":"eng","text":"We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. "}],"date_created":"2026-03-02T10:09:23Z","license":"https://creativecommons.org/licenses/by/4.0/","OA_place":"publisher","OA_type":"hybrid","author":[{"first_name":"Victor","orcid":"0000-0002-0704-7026","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","last_name":"Wang","full_name":"Wang, Victor"},{"first_name":"Max","full_name":"Xu, Max","last_name":"Xu"}],"supplementarymaterial":"no","main_file_link":[{"url":"https://doi.org/10.1017/prm.2026.10123","open_access":"1"}],"researchdata_availability":"no","ddc":["510"],"arxiv":1,"project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"month":"01","oa":1,"page":"1-15","status":"public","title":"Average sizes of mixed character sums","quality_controlled":"1","corr_author":"1","date_updated":"2026-07-16T08:39:57Z","_id":"21385","date_published":"2026-01-01T00:00:00Z","oa_version":"Published Version","PlanS_conform":"1","external_id":{"arxiv":["2411.14181"]},"publication":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","citation":{"chicago":"Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press, 2026. <a href=\"https://doi.org/10.1017/prm.2026.10123\">https://doi.org/10.1017/prm.2026.10123</a>.","short":"V. Wang, M. Xu, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2026) 1–15.","apa":"Wang, V., &#38; Xu, M. (2026). Average sizes of mixed character sums. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2026.10123\">https://doi.org/10.1017/prm.2026.10123</a>","ama":"Wang V, Xu M. Average sizes of mixed character sums. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. 2026:1-15. doi:<a href=\"https://doi.org/10.1017/prm.2026.10123\">10.1017/prm.2026.10123</a>","ieee":"V. Wang and M. Xu, “Average sizes of mixed character sums,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. Cambridge University Press, pp. 1–15, 2026.","mla":"Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, Cambridge University Press, 2026, pp. 1–15, doi:<a href=\"https://doi.org/10.1017/prm.2026.10123\">10.1017/prm.2026.10123</a>.","ista":"Wang V, Xu M. 2026. Average sizes of mixed character sums. Proceedings of the Royal Society of Edinburgh: Section A Mathematics., 1–15."},"publisher":"Cambridge University Press","doi":"10.1017/prm.2026.10123","ec_funded":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"epub_ahead","language":[{"iso":"eng"}],"year":"2026","department":[{"_id":"TiBr"}],"type":"journal_article"}]