  We study asymptotic relations connecting unipotent averages of
$Sp(2g,\mathbb{Z})$ automorphic forms to their integrals over the moduli space
of principally polarized abelian varieties. We obtain reformulations of the
Riemann hypothesis as a class of problems concerning the computation of the
equidistribution convergence rate in those asymptotic relations. We discuss
applications of our results to closed string amplitudes. Remarkably, the
Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring
in perturbative closed string theory.
