  In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick
established relationships of the mean-square of sums of the divisor function
$d_k(f)$ over short intervals and over arithmetic progressions for the function
field $\mathbb F_q[T]$ 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.
