Regularized Infinite Products

• Regularized Infinite Products are methods that assign finite, canonical values to divergent infinite products through analytic continuation and zeta-regularization.
• They employ analytic continuation of zeta and L-functions to regularize products over natural numbers, primes, and quadratic integer rings, yielding explicit evaluations via the Gamma function.
• These techniques have broad applications in number theory, quantum field theory, and spectral analysis, highlighting deep structural analogies across arithmetic settings.

A regularized infinite product is an assignment of a finite, often canonical, value to an infinite product that formally diverges, achieved by analytic or functional extension, typically via zeta-regularization, analytic continuation, or structural summation methods. Regularization is essential in analytic number theory, quantum field theory, and mathematical physics, particularly for extracting meaningful invariants from infinite combinatorial or arithmetic data.

1. Classical Regularization of Infinite Products

The archetype of regularization is zeta-regularization, where the divergent product

$\prod_{n \geq 1} n$

is assigned the finite value $\sqrt{2\pi}$, motivated by the functional equation for the Riemann zeta function and the analytic value $\zeta'(0)$. Similarly, the regularized product over all primes, as introduced by Muñoz Garcia and Pérez-Marco, is

$\prod_{p} p = 4\pi^2,$

with the relationship $4\pi^2 = (\sqrt{2\pi})^4$. Analytically, these products are evaluated via

$P = \exp(-\zeta'(0)),$

where $\zeta(s)$ is the appropriate zeta function (e.g., the Riemann zeta function for the natural numbers) and the prime product invokes either the zeta function's relation to prime distribution or the explicit prime zeta function.

Divergences are circumvented by analytic continuation to arguments (typically $s=0$) where the formal product is ill-defined, utilizing the functional identities and special values of zeta and $L$-functions.

2. Regularized Products in Imaginary Quadratic Integer Rings

2.1 Gauss Integers and Primes

For the Gaussian integers $\mathbb{Z}[i]$, a regularized infinite product can be formed over all nonzero elements, up to units and orbits, i.e.,

$P_4^2 = \prod_{(a,b) \ne (0,0)} (a^2 + b^2),$

which is regularized via the Dedekind zeta function for $\mathbb{Q}(i)$, denoted $\zeta_4(s)$,

$$\zeta_4(s) = \sum_{(a,b)\ne (0,0)} (a^2 + b^2)^{-s}.$$

Zeta-regularization assigns

$P_4^2 = \exp\left[ -\zeta_4'(0) \right].$

Using properties of Dirichlet $L$-functions, this yields the explicit formula

$$P_4 = \frac{\Gamma\left(\frac{1}{4}\right)^2}{2\sqrt{\pi}} \approx 3.708149.$$

For the Gaussian primes (irreducibles in $\mathbb{Z}[i]$), the regularized product utilizes a "prime zeta function" $Z_4(s)$, yielding

$$Q_4^2 = \prod_{\text{Gauss primes}} N(z) = \exp[-Z_4'(0)],$$

and the computation establishes

$Q_4 = P_4^8,$

so that

$$\prod_{\text{Gauss primes } z} z = \left( \frac{\Gamma(\frac{1}{4})^2}{2\sqrt{\pi}} \right)^8.$$

These evaluations employ analytic continuation, decomposition of Dedekind zeta functions into Dirichlet $L$-functions, Hurwitz zeta values, and tight combinatorial accounting of the symmetries (units and orbits under group actions) intrinsic to $\mathbb{Z}[i]$.

2.2 Eisenstein Integers and Primes

For Eisenstein integers $\mathbb{Z}[\omega]$ ($\omega = e^{2\pi i/3}$), the regularized product over nonzero integers considers

$P_3^2 = \prod_{(a,b)\ne (0,0)}(a^2 - a b + b^2),$

regularized by the Dedekind zeta function $\zeta_3(s)$: $$P_3^2 = \exp\left[ -\zeta_3'(0) \right], \quad \text{with}\quad P_3 = \frac{3^{1/4}\Gamma(\frac{1}{3})^3}{2\pi} \approx 4.027065.$$ For Eisenstein primes,

$Q_3 = P_3^{12},$

so that

$$\prod_{\text{Eisenstein primes } z} z = \left( \frac{3^{1/4}\Gamma(\frac{1}{3})^3}{2\pi} \right)^{12}.$$

Again, central to the computation is decomposition into Dirichlet $L$-functions and careful regularization via analytic continuation.

3. Analytic and Zeta-Regularization Methodologies

The general regularization procedure is as follows:

1. Express the formal infinite product as

$\prod_{n} a_n,$

often divergent.

2. Define an associated (generalized) zeta function

$\zeta(s) = \sum_{n} a_n^{-s}$

or a logarithmic derivative for products over primes.

3. Obtain the regularized value as

$\exp(-\zeta'(0))$

after analytic continuation and (where applicable) explicit evaluation using special functions such as the Gamma, Hurwitz, and Dirichlet $L$-functions.

4. If the product is over primes, a “prime zeta function” is constructed, careful analytic treatment ensures convergence, and the value is regularized in the same fashion.

Potential obstructions arise for prime zeta functions, whose natural boundaries restrict direct analytic continuation. However, for Dedekind zeta functions in quadratic fields, reduction to sums and products of classical zeta and $L$-functions allows for explicit and rigorous evaluation.

4. Structural Universality and Analogies

There is a remarkable analogy between the regularized products for the classical integers and those for imaginary quadratic integer rings:

• For rational integers: $\prod_{p} p = (\prod_{n \ge 1} n)^4$.
• For Gauss integers: $Q_4 = P_4^8$.
• For Eisenstein integers: $Q_3 = P_3^{12}$.

A plausible implication is that for a given ring of integers of quadratic imaginary field (with $d$-th roots of unity), the regularized product over irreducibles (primes), $Q_d$, is related to the product over all nonzero elements, $P_d$, via

$Q_d = P_d^{2\phi(d)},$

where $\phi$ is Euler’s totient function, corresponding to the number of units. This structural relation encodes the enhanced symmetry and density of the various rings and primes under unit group actions.

5. Explicit Results for Gauss and Eisenstein Integer Rings

The following table summarizes all explicit values established by zeta-regularization in this context:

Object	Regularized Product	Explicit Value
All natural numbers	$P = \sqrt{2\pi}$
All natural primes	$Q = 4\pi^2$	$Q = P^4$
All nonzero Gauss integers	$P_4 = \frac{\Gamma(\frac{1}{4})^2}{2\sqrt{\pi}}$	$\approx 3.708149$
All Gauss primes	$Q_4 = P_4^8$	$\approx 7.0116 \times 10^5$
All nonzero Eisenstein integers	$P_3 = \frac{3^{1/4}\Gamma(\frac{1}{3})^3}{2\pi}$	$\approx 4.027065$
All Eisenstein primes	$Q_3 = P_3^{12}$	$\approx 2.3978 \times 10^7$

All values are rigorously obtained by evaluating the appropriate Dedekind and $L$-functions and applying analytic continuation to $s=0$.

6. Broader Significance and Applications

The methodology of regularizing infinite products over arithmetic or geometric data using analytic continuation of zeta or $L$-functions is not limited to classical number theory. The technique underpins approaches to determinants of Laplacians (spectral invariants), quantum partition functions, and evaluation of Feynman path integrals in quantum field theory. The explicit connection of regularized products in the rings $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, along with the structural formulae involving the Gamma function, demonstrate the deep interrelation between spectral, algebraic, and combinatorial invariants.

The use of special function evaluations, particularly those involving the Gamma and Hurwitz zeta functions (as in the explicit expressions for $P_4$ and $P_3$), is essential for computation in related analytic and physical contexts.

7. Summary

Regularized infinite products, typified by zeta-regularization and analytic continuation, consistently assign finite, canonical values to divergent products over intricate algebraic structures. The explicit evaluations for the rings of integers in $\mathbb{Q}(i)$ and $\mathbb{Q}(\omega)$, and their sets of irreducibles, establish universal structural relations paralleling classical results for $\mathbb{Z}$ and its primes. The techniques deployed—analytic continuation, decomposition of Dedekind zeta functions, and explicit account of unit group actions—illuminate a deep arithmetic and analytic structure that extends broadly across number theory and mathematical physics.