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

Artur Kawalec

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.

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

Abstract

We analytically continue the Euler prime product for (except for its pole ) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at 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.

Paper Structure

This paper contains 5 sections, 1 theorem, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. A note on Euler product at $s=1$
  4. A note on other related Euler Products
  5. Numerical computation

Key Result

Theorem 1

valid for $\Re(s)>\tfrac{1}{2}$ and $(s\neq 1)$ assuming (RH).

Figures (4)

  • Figure 1: A plot of the analytically continued Euler product for real $s$ for $x=10^6$
  • Figure 2: A plot of the analytically continued Euler product at $\sigma=0.8$ for $x=10^3$
  • Figure 3: A plot of the analytically continued Euler product at $\sigma=0.55$ for $x=10^3$
  • Figure 4: A plot of the analytically continued Euler product at $\sigma=0.55$ for $x=10^5$

Theorems & Definitions (1)

  • Theorem 1