A general inversion formula for summatory arithmetic functions and its application to the summatory function of the Moebius function

We prove an inversion formula for summatory arithmetic functions. As an application, we obtain an arithmetic relationship between summatory Piltz divisor functions and a sum of the M\"obius function over certain integers, denoted by $M(x,y)$. With this relationship, using bounds for the main and remainder terms in the $k$-divisor problems we deduce conditional and unconditional results concerning $M(x,y)$ and the zero-free region of the Riemann zeta-function and Dirichlet $L$-functions.