On the abscissae of Weil representation zeta functions for procyclic groups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℤ</m:mi> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>∣</m:mo> <m:mi>p</m:mi> <m:mo>∉</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {Z[p⁻¹ p∈ S]} . Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>β</m:mi> <m:mo>≤</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {1≤β≤ 2} is the Weil abscissa of some procyclic group.