The $\zeta$-regularized product over all primes

TL;DR

This paper proves that the $zeta$-regularized product over all primes equals $pi e^{mu}$, linking it to the non-trivial zeros of the Riemann zeta function, revealing a deep connection between prime products and zeta zeros.

Contribution

It establishes a precise formula for the $zeta$-regularized product over all primes involving the non-trivial zeros of the zeta function, a novel analytical result.

Findings

The $zeta$-regularized product over all primes equals $pi e^{mu}$.

The constant $mu$ is related to the non-trivial zeros of $zeta(s)$.

Provides a new link between prime products and the zeros of the zeta function.

Abstract

In this paper we prove that the $\zeta$-regularized product over all primes is $\pi e^{\mu}$, where $\mu$ is closely related with the non-trivial zeros of the $\zeta(s)$.