On the abscissae of Weil representation zeta functions for procyclic groups

TL;DR

This paper proves that for random procyclic groups, the abscissa of convergence of their Weil representation zeta functions can be any real number between 1 and 2, unconditionally confirming conjectured dependencies.

Contribution

It establishes that the abscissa of convergence depends only on prime divisors and matches Dedekind zeta functions for procyclic groups, unconditionally for random models.

Findings

Any real number between 1 and 2 can be realized as the Weil abscissa of some procyclic group.

The conjectured dependence on prime divisors holds unconditionally in the random group model.

The abscissa matches that of the Dedekind zeta function for the set of primes dividing the group order.

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.