{"article_processing_charge":"No","issue":"10","page":"323-370","month":"03","year":"2025","quality_controlled":"1","volume":12,"language":[{"iso":"eng"}],"type":"journal_article","publisher":"American Mathematical Society","publication_status":"published","intvolume":"        12","doi":"10.1090/btran/186","scopus_import":"1","abstract":[{"lang":"eng","text":"In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747],\r\nthe authors studied the mean-square of certain sums of the divisor function\r\ndk(f) over the function field Fq[T] in the limit as q →∞ and related these\r\nsums to integrals over the ensemble of symplectic matrices, along similar lines\r\nas previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z.\r\n288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue\r\na more detailed study of both the symplectic and orthogonal matrix integrals,\r\nrelating them to symmetric function theory. The function field results lead to\r\nconjectures concerning analogous questions over number fields."}],"citation":{"ama":"Kuperberg VZ, Lalín M. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. <i>Transactions of the American Mathematical Society, Series B</i>. 2025;12(10):323-370. doi:<a href=\"https://doi.org/10.1090/btran/186\">10.1090/btran/186</a>","short":"V.Z. Kuperberg, M. Lalín, Transactions of the American Mathematical Society, Series B 12 (2025) 323–370.","chicago":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” <i>Transactions of the American Mathematical Society, Series B</i>. American Mathematical Society, 2025. <a href=\"https://doi.org/10.1090/btran/186\">https://doi.org/10.1090/btran/186</a>.","ieee":"V. Z. Kuperberg and M. Lalín, “Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime,” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 12, no. 10. American Mathematical Society, pp. 323–370, 2025.","ista":"Kuperberg VZ, Lalín M. 2025. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. Transactions of the American Mathematical Society, Series B. 12(10), 323–370.","apa":"Kuperberg, V. Z., &#38; Lalín, M. (2025). Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. <i>Transactions of the American Mathematical Society, Series B</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/btran/186\">https://doi.org/10.1090/btran/186</a>","mla":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 12, no. 10, American Mathematical Society, 2025, pp. 323–70, doi:<a href=\"https://doi.org/10.1090/btran/186\">10.1090/btran/186</a>."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.04969"}],"author":[{"last_name":"Kuperberg","first_name":"Vivian Zieve","full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"},{"last_name":"Lalín","first_name":"Matilde","full_name":"Lalín, Matilde"}],"title":"Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime","day":"20","OA_place":"repository","article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-07-14T11:24:55Z","status":"public","mathsc":["11N60","05A15","11M50","11N56"],"_id":"22200","oa_version":"Preprint","date_created":"2026-06-29T12:59:46Z","publication":"Transactions of the American Mathematical Society, Series B","extern":"1","OA_type":"green","publication_identifier":{"issn":["2330-0000"]},"arxiv":1,"date_published":"2025-03-20T00:00:00Z","external_id":{"arxiv":["2212.04969"]},"oa":1}