Regularized products of Gauss and Eisenstein integers and primes

TL;DR

This paper heuristically computes regularized infinite products over Gauss and Eisenstein integers and primes, providing explicit formulas inspired by prior work on prime products related to $4\\pi^2$, advancing understanding of these complex number systems.

Contribution

It introduces heuristic methods to evaluate regularized infinite products over Gaussian and Eisenstein primes, offering explicit expressions inspired by classical prime product evaluations.

Findings

Explicit formulas for regularized products over Gaussian and Eisenstein integers.

Heuristic approach inspired by prime product evaluations related to $4\pi^2$.

Connections to classical prime product formulas.

Abstract

We provide heuristic computations \`a la Euler of the regularized infinite products of Gauss and Eisenstein integers and primes. Our approach, yielding explicit expressions, is inspired by the work by Mu\~noz Garc\'ia and P\'erez-Marco, who evaluated the product of all natural primes to $4\pi^2$.