Abstract:Euler regularized the divergent product of all natural numbers and found beautiful formulas for regularized sums of integer powers of natural numbers. These derivations essentially relied on what is now called the zeta-regularization technique, although analytical continuation had not yet been invented. This classic method is however not applicable to the product of all primes, as the prime zeta function has a natural boundary along the imaginary axis. Muñoz García and Pérez-Marco overcame this obstacle and evaluated the product of all primes to $4\pi^2$ by finding an appropriately regularized value of the derivative of the prime zeta function at the origin, lying on the natural boundary. We extend their approach in two novel directions. First, we show how to make sense of the sum of all primes. This regularization requires going a finite distance beyond the natural boundary. Second, we determine the regularized products of integers and primes in the nine imaginary quadratic fields where integers have a unique factorization into primes, and establish a general power-law relationship between products of integers and primes. Two well-known examples are Gauss and Eisenstein integers. The interest in this approach goes beyond number theory. In a variety of physical situations, the zeta-regularization technique is indeed not applicable because the relevant zeta function has a natural boundary.
From: Jean-Marc Luck [view email]
[v1]
Tue, 23 Jun 2026 13:04:44 UTC (54 KB)
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