Regularized products of Gauss and Eisenstein integers and primes

TL;DR

The paper develops Euler-like, zeta-regularized infinite products for Gauss and Eisenstein integers and their primes. By expressing the regularized products through Dedekind zeta and Dirichlet L-functions, the authors derive explicit closed forms in terms of Gamma functions: $|P_4|=\frac{\Gamma(1/4)^2}{2\sqrt{\pi}}$ and $|P_3|=\frac{3^{1/4}\Gamma(1/3)^3}{2\pi}$, with prime-product magnitudes obeying $|\Pi_4|=|P_4|^8$ and $|\Pi_3|=|P_3|^{12}$. These results generalize the classical Euler product regularization for natural numbers, suggesting a deeper structure linking algebraic integer rings, their unit groups, and zeta/L-function factorization. The approach highlights connections to quantum billiards and integrable spectra, and points to a broader, as yet uncovered, underlying property across number fields. Overall, the work extends a celebrated prime-product result to higher-dimensional integer rings, providing explicit, testable expressions via well-studied special functions.

Abstract

We provide heuristic computations à 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 Muñoz García and Pérez-Marco, who evaluated the product of all natural primes to $4π^2$.