On the abscissae of Weil representation zeta functions for procyclic groups

Abstract

A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq \beta \leq 2$ is the Weil abscissa of some procyclic group.