Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime Kuperberg, Vivian Zieve Lalín, Matilde In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747], the authors studied the mean-square of certain sums of the divisor function dk(f) over the function field Fq[T] in the limit as q →∞ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z. 288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields. American Mathematical Society 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/22200 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1090/btran/186 info:eu-repo/semantics/altIdentifier/issn/2330-0000 info:eu-repo/semantics/altIdentifier/arxiv/2212.04969 info:eu-repo/semantics/openAccess