Regularized products of Gauss and Eisenstein integers and primes

Technical Summary: Regularized Products of Gauss and Eisenstein Integers and Primes

Problem Statement
Infinite products of integers and primes are inherently divergent. While the zeta-regularization technique (rooted in Euler's work and formalized via analytic continuation) allows for the assignment of finite values to such products, explicit evaluations for complex integer rings remain less explored than for the natural integers. Specifically, this paper addresses the calculation of regularized infinite products for the rings of Gauss integers ($\mathbb{Z}[i]$) and Eisenstein integers ($\mathbb{Z}[\omega]$), where $\omega = e^{2\pi i/3}$. The authors aim to determine the regularized products of all non-zero elements in these rings ($P_4$ and $P_3$) and the products of their associated primes ($\Pi_4$ and $\Pi_3$).

Methodology
The authors employ a heuristic approach "à la Euler," inspired by the work of Mu˜noz Garc´ıa and P´erez-Marco [14, 15], who evaluated the product of all natural primes to $4\pi^2$. The methodology proceeds as follows:

1. Zeta-Regularization Framework: The divergent product $P$ is formally expressed as $P = \exp(-\zeta'(0))$, where $\zeta(s)$ is the relevant zeta function. For products of primes, the prime zeta function $Z(s)$ is utilized, with the product given by $\Pi = \exp(-Z'(0))$.
2. Handling Symmetry and Phase: Since the rings $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$ possess non-trivial unit groups (4 units for Gauss, 6 for Eisenstein), the phase of the infinite product is ill-defined due to rotational invariance. Consequently, the authors focus on the modulus of the products, specifically $|P|^2$ and $|\Pi|^2$, which correspond to products of the squared norms of the integers and primes.
3. Dedekind Zeta Functions: The regularization relies on the Dedekind zeta functions $\zeta_4(s)$ for $\mathbb{Z}[i]$ and $\zeta_3(s)$ for $\mathbb{Z}[\omega]$. These are decomposed into the Riemann zeta function $\zeta(s)$ and specific Dirichlet $L$-functions ($L_4(s)$ and $L_3(s)$).
4. Prime Decomposition Analysis: To evaluate the prime products, the authors analyze the factorization of rational primes within these rings (inert, split, and ramified cases). They construct the prime zeta functions $Z_4(s)$ and $Z_3(s)$ by summing over the norms of the primes, weighted by their multiplicities (the number of distinct prime elements associated with a given norm).
5. Analytic Continuation: The authors utilize known values and derivatives of the Riemann zeta function, Hurwitz zeta functions, and Dirichlet $L$-functions at $s=0$ (e.g., $\zeta(0) = -1/2$, $\zeta'(0) = -\frac{1}{2}\ln(2\pi)$, and Lerch's formula for derivatives of Hurwitz zeta functions) to compute the final regularized values.

Key Contributions and Results

• Gauss Integers ($\mathbb{Z}[i]$):

  • The regularized product of all non-zero Gauss integers, $P_4$, yields a modulus:
    $$|P_4| = \frac{\Gamma(1/4)^2}{2\sqrt{\pi}} \approx 3.708149$$
  • The regularized product of all Gauss primes, $\Pi_4$, satisfies the relation:
    $$|\Pi_4| = |P_4|^8$$

• Eisenstein Integers ($\mathbb{Z}[\omega]$):

  • The regularized product of all non-zero Eisenstein integers, $P_3$, yields a modulus:
    $$|P_3| = \frac{3^{1/4} \Gamma(1/3)^3}{2\pi} \approx 4.027065$$
  • The regularized product of all Eisenstein primes, $\Pi_3$, satisfies the relation:
    $$|\Pi_3| = |P_3|^{12}$$

• Structural Analogies:
  The paper highlights a striking pattern connecting these results to the product of natural integers ($P = \sqrt{2\pi}$) and natural primes ($\Pi = 4\pi^2$). Specifically, the relationship $\Pi = P^4$ for natural numbers is generalized to $|\Pi_4| = |P_4|^8$ for Gauss integers and $|\Pi_3| = |P_3|^{12}$ for Eisenstein integers.

Significance and Claims
The paper positions itself as a direct extension of the work by Mu˜noz Garc´ıa and P´erez-Marco [14, 15], applying their heuristic regularization techniques to the simplest generalizations of natural integers: the Gaussian and Eisenstein integers. The authors note that these rings correspond to the spectra of the simplest integrable quantum billiards (the square and the equilateral triangle).

The primary significance claimed is the derivation of explicit, closed-form expressions for these regularized products involving Gamma functions. The authors observe the consistent power-law relationship between the product of integers and the product of primes across these different number fields (exponents 4, 8, and 12). However, they explicitly state that the underlying reason for this "striking resemblance" and the existence of a more general property governing these exponents is currently unknown to them. The work remains a mathematical exploration of regularization in algebraic number fields without proposing new physical applications or future experimental directions.