Zeta-regularization and natural boundaries: Sums and products of integers and primes

Technical Summary: Zeta-Regularization and Natural Boundaries

Problem Statement
The paper addresses the challenge of assigning finite, regularized values to divergent infinite products and sums involving integers and prime numbers. While Euler successfully regularized the product of all natural numbers ($P = \sqrt{2\pi}$) and sums of integer powers using the Riemann zeta function $\zeta(s)$, extending these methods to the product of all natural primes ($\Pi$) presents a fundamental obstacle. The prime zeta function, $P(s) = \sum p^{-s}$, possesses a natural boundary along the imaginary axis, rendering it non-analytic in the neighborhood of the origin ($s=0$). Consequently, the standard zeta-regularization technique, which relies on evaluating derivatives like $P'(0)$, fails for the product of primes. Furthermore, the paper tackles the even more difficult problem of regularizing the sum of all primes, $P(-1)$, which requires evaluating the function at a point ($s=-1$) located a finite distance beyond this natural boundary. Finally, the authors seek to generalize these results to the nine imaginary quadratic fields where integers admit unique factorization into primes.

Methodology
The authors employ a combination of analytic continuation, Möbius inversion, and heuristic regularization schemes adapted from the work of Muñoz García and Pérez-Marco.

1. Handling the Natural Boundary: For the product of primes, the authors utilize the Artin-Hasse identity to express the prime zeta function $P(s)$ in terms of the Riemann zeta function $\zeta(s)$ via the relation $P(s) = \sum_{n \ge 1} \frac{\mu(n)}{n} \log \zeta(ns)$. This allows the evaluation of $P'(0)$ by regularizing the resulting sums involving the Möbius function $\mu(n)$ using the identity $\sum \mu(n)n^{-s} = 1/\zeta(s)$.
2. Sum of Primes: To compute the sum of primes $P(-1)$, the authors analyze the series expansion of $P(-1+\epsilon)$ near the natural boundary. By separating odd and even terms in the Möbius sum and utilizing the functional equation (reflection formula) of the Riemann zeta function, they isolate divergent logarithmic terms. These divergences are regularized using known values of Dirichlet series involving $\mu(n)$, leaving a convergent series that yields the final finite value.
3. Extension to Quadratic Fields: The methodology is extended to the nine imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ with class number one (Heegner numbers). The authors utilize Dedekind zeta functions $\zeta_\Delta(s)$ and associated Dirichlet L-functions $L_\Delta(s)$. They derive the prime zeta functions for these fields ($P_\Delta(s)$) by analyzing the splitting, inert, and ramified behavior of natural primes within the rings of integers (Gauss integers $\mathbb{Z}[i]$, Eisenstein integers $\mathbb{Z}[\omega]$, and others). The regularized products are determined by relating the derivatives of the prime zeta functions to the derivatives of the Dedekind zeta functions at $s=0$.

Key Contributions and Results

• Regularized Sum of Primes: The paper provides a heuristic derivation for the sum of all natural primes, a value previously considered inaccessible due to the natural boundary. The result is:
  $$\sum_{p} p = P(-1) = \frac{5}{2} + \sum_{n \ge 1} \frac{\mu(n)}{n} \log \left( \frac{n! (e/n)^n \zeta(n+1)}{\sqrt{2\pi n}} \right) \approx 2.925292456\dots$$
  The non-trivial regularization yields the rational part $5/2$, while the remaining series is convergent.

• Power-Law Relations: The authors establish a universal power-law relationship between the regularized product of all integers ($P_\Delta$) and the regularized product of all primes ($\Pi_\Delta$) in these number fields. The relationship is given by:
  $$|\Pi_\Delta| = |P_\Delta|^{2w_\Delta}$$
  where $w_\Delta$ is the number of units in the ring of integers.

  • For natural integers ($\mathbb{Z}$): $w=2$ (units $\pm 1$), yielding $\Pi = P^4$.
  • For Eisenstein integers ($\mathbb{Z}[\omega]$): $w=6$, yielding $|\Pi_3| = |P_3|^{12}$.
  • For Gauss integers ($\mathbb{Z}[i]$): $w=4$, yielding $|\Pi_4| = |P_4|^8$.
  • For the other seven imaginary quadratic fields: $w=2$, yielding $|\Pi_\Delta| = |P_\Delta|^4$.

• Explicit Values: The paper computes explicit numerical values and closed-form expressions (involving Gamma functions and Chowla-Selberg gamma ratios) for the regularized products of integers in all nine imaginary quadratic fields. For example, $|P_3|^2 = \frac{2\pi}{3} \frac{\Gamma(1/3)^3}{\Gamma(2/3)^3}$ and $|P_4|^2 = \frac{\pi}{2} \frac{\Gamma(1/4)^2}{\Gamma(3/4)^2}$.

Significance and Scope
The paper claims that these derivations extend the reach of zeta-regularization techniques to situations where the relevant zeta functions possess natural boundaries, a scenario common in various physical contexts such as quantum mechanics in specific potentials, statistical mechanics, and gauge theories. The authors note that while the derivation of the prime product formula (1.2) was a recent breakthrough, the calculation of the prime sum (1.6) represents a novel extension of these techniques "beyond the natural boundary."

The work is presented primarily within a number-theoretic context. The authors explicitly state that the interest in this approach extends to physics because "in a variety of physical situations, the zeta-regularization technique is indeed not applicable because the relevant zeta function has a natural boundary." However, the paper does not propose specific new physical experiments or claim that these specific number-theoretic results solve existing physical problems; rather, it offers mathematical tools and analogues that may be applicable to such problems. The paper concludes by noting that the validity of the universal formula (1.10) for more general algebraic number fields, particularly real quadratic fields, remains an open question.