The Dirichlet series that generates the Möbius function is the inverse of the Riemann zeta function in the right half of the critical strip

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.