Sums of divisor functions and von Mangoldt convolutions in 𝔽 q [T] leading to symplectic distributions

Abstract

Abstract In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽 q ⁢ [ t ] \mathbb{F}_{q}[t] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function d k ⁢ ( f ) {d_{k}(f)} over short intervals and over arithmetic progressions for the function field 𝔽 q ⁢ [ T ] {\mathbb{F}_{q}[T]} to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.