Möbius function and primes: an identity factory with applications

We investigate the sums ∑ n ≤ X , ( n , q ) = 1 μ ( n ) n s log k ⁡ ( X n ) \sum _{n\le X, (n,q)=1}\frac {\mu (n)}{n^{s}}\log ^{k}\left (\frac {X}{n}\right ) , where k ∈ { 0 , 1 } k\in \{0,1\} , s ∈ C s\in \mathbb {C} , ℜ s > 0 \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 ∑ n ≤ X Λ ( n ) / n ≤ log ⁡ X \sum _{n\le X}\Lambda (n)/n\le \log X , valid for any X ≥ 1 X\geq 1 .