Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)

TL;DR

This paper presents a method to analytically continue the Euler prime product for complex numbers with real part greater than 1/2, assuming the Riemann Hypothesis, and demonstrates numerical verification using Pari/GP.

Contribution

It introduces a new factor to extend the Euler prime product beyond its original domain under RH and applies this technique to related Euler products.

Findings

Successfully analytically continued the Euler prime product assuming RH.

Provided a Pari/GP script for numerical verification.

Connected the continuation method to Mertens's 3rd Theorem.

Abstract

We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case, and how to apply the same technique to analytically continue other similar Euler products. In the last part, we also construct a simple script in Pari/GP to compute the Euler product and verify the calculations numerically.