Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups
Martin Jann, Steffen Kionke
TL;DR
The paper determines the Weil abscissa $\alpha(H_S)$ for three procyclic groups $H_S=\prod_{p\in S} \mathbb{Z}_p$ where $S$ consists of primes in arithmetic progressions: $p\equiv1\bmod3$, $p\equiv1\bmod4$, and $p\equiv1,3\bmod8$. It leverages the link between primes in these progressions and representations of integers by binary quadratic forms $f_n(x,y)=x^2+ny^2$ with $n\in\{1,2,3\}$, together with Iwaniec's theorem to construct minorants for the Weil representation zeta function. By producing divergent minorants for $\ln \zeta_{H_S}^W(2-\varepsilon)$, the authors establish the lower bound $\alpha(H_S)\ge 2$, which combined with the general upper bound yields $\alpha(H_S)=2$ for each case. This validates a case of Conjecture A for arithmetic-progression prime sets and demonstrates a method potentially extendable to unions of progressions in the context of Weil's zeta framework.
Abstract
Here we show that the Weil abscissa of the procyclic groups $\prod_{p \in S} \mathbb{Z}_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.