The Dirichlet series that generates the M\"obius function is the inverse of the Riemann zeta function in the right half of the critical strip

TL;DR

This paper introduces a new criterion for the Riemann hypothesis and proves that the Dirichlet series generating the Möbius function converges for real parts greater than 1/2, using a novel step function approach.

Contribution

It presents a new convergence result for the Möbius function's Dirichlet series in the critical strip, linked to a novel criterion for the Riemann hypothesis.

Findings

Proves convergence of the Möbius series for Re(s) > 1/2

Introduces a step function approach to analyze the Dirichlet eta function

Provides a new criterion related to the Riemann hypothesis

Abstract

In this paper I introduce a criterion for the Riemann hypothesis, and then using that I prove $\sum_{k=1}^\infty \mu(k)/k^s$ converges for $\Re(s) > \frac{1}{2}$. I use a step function $\nu(x) = 2\{x/2\} - \{x\}$ for the Dirichlet eta function ($\{x\}$ is the fractional part of $x$), which was at the core of my investigations, and hence derive the stated result subsequently.