Möbius function, an identity factory with applications

By using an identity relating a sum to an integral, we obtain a family of identities for the averages $M(X)=\sum_{n\leq X} \mu(n)$ and $m(X)=\sum_{n\leq X} \mu(n)/n$. Further, by choosing some specific families, we study two summatory functions related to the Möbius function, $\mu(n)$, namely $\sum_{n\leq X} \mu(n)/n^s$ and $\sum{n\leq X} \mu(n)/n^s \log(X/n)$, where $s$ is a complex number and $\Re s >0$. We also explore some applications and examples when s is real. (joint work with O. Ramaré)