Möbius function and primes: an identity factory with applications

We investigate the sums ${\sum }_{n\le X,(n,q)=1}\frac{\mu (n)}{n^s}{\log }^k\left(\frac{X}{n}\right)$, where $k\in \{0,1\}$, $s\in \mathbb{C}$, $\mathrm{\Re }s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality ${\sum }_{n\le X}\mathrm{\Lambda }(n)\mathrm{/}n\le \log X$, valid for any $X\ge 1$.