On the prime zeta function and the Riemann hypothesis

TL;DR

The paper explores the prime zeta function's connection to the Riemann hypothesis, initially aiming to prove it but ultimately recognizing a fundamental flaw in the proposed proof.

Contribution

It analyzes the prime zeta function's properties and discusses the challenges in proving the Riemann hypothesis using this approach.

Findings

The proof of Theorem 3 is flawed.

The prime zeta function's properties relate to the Riemann hypothesis.

Classical tools like bounds and oscillation theorems are used in the analysis.

Abstract

By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of "Theorem 3" is fundamentally flawed. The main tools of our argument are: bounds and oscillation theorems for the prime counting function, classical properties of Dirichlet series and the identity theorem for real-analytic functions.