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

Euler regularized the divergent product of all positive integers and found beautiful formulas for regularized sums of integer powers of positive integers. 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 Garc\'ia and P\'erez-Marco overcame this obstacle and evaluated the product of all primes to 4π² by finding an appropriately regularized value of the derivative of the prime zeta function at the origin, lying on the natural boundary. Their approach heavily relies on the M\"obius function and could therefore be called a M\"obius regularization. We extend this 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.