Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

article Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime published yes Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 Matilde Lalín author 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 Society2025 eng Transactions of the American Mathematical Society, Series B 2330-0000 2212.0496910.1090/btran/186 1210323-370 yes Kuperberg VZ, Lalín M. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. Transactions of the American Mathematical Society, Series B. 2025;12(10):323-370. doi:<a href="https://doi.org/10.1090/btran/186">10.1090/btran/186</a> V.Z. Kuperberg, M. Lalín, Transactions of the American Mathematical Society, Series B 12 (2025) 323–370. V. Z. Kuperberg and M. Lalín, “Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime,” Transactions of the American Mathematical Society, Series B, vol. 12, no. 10. American Mathematical Society, pp. 323–370, 2025. 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. Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” Transactions of the American Mathematical Society, Series B. American Mathematical Society, 2025. <a href="https://doi.org/10.1090/btran/186">https://doi.org/10.1090/btran/186</a>. Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” Transactions of the American Mathematical Society, Series B, 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>. Kuperberg, V. Z., & Lalín, M. (2025). Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. Transactions of the American Mathematical Society, Series B. American Mathematical Society. <a href="https://doi.org/10.1090/btran/186">https://doi.org/10.1090/btran/186</a> 222002026-06-29T12:59:46Z2026-07-14T11:24:55Z