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Analytical continuation of prime zeta function for $\Re(s)>\frac{1}{2}$ assuming (RH)

Artur Kawalec

Abstract

We derive a simple expression to analytically continue the prime zeta function to the domain $\Re(s)>\frac{1}{2}$ assuming (RH) and taking into account a proper branch cut. We also verify the formula numerically and provide several plots.

Analytical continuation of prime zeta function for $\Re(s)>\frac{1}{2}$ assuming (RH)

Abstract

We derive a simple expression to analytically continue the prime zeta function to the domain assuming (RH) and taking into account a proper branch cut. We also verify the formula numerically and provide several plots.

Paper Structure

This paper contains 3 sections, 1 theorem, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Numerical Computation

Key Result

Theorem 1

which is analytic for $\Re(s)>\tfrac{1}{2}$ with a branch cut on $(\tfrac{1}{2},1]$ assuming (RH).

Figures (3)

  • Figure 1: A plot of $\Re[P(s)]$ by equation (3) for $\Re(s)$ variable for limit variable $x=10^4$
  • Figure 2: A plot of $\Re[P(\sigma+it)]$ by equation (3) for at $\sigma=0.75$ and vertical line $t=0.1$ to $t=50$ for limit variable $x=10^4$
  • Figure 3: A plot of $\Im[P(\sigma+it)]$ by equation (3) for at $\sigma=0.75$ and vertical line $t=0.1$ to $t=50$ for limit variable $x=10^4$

Theorems & Definitions (1)

  • Theorem 1