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 ) ∑ n≤ X, (n,q)=1 {μ (n)}{nˢ}log ᵏ (X/n ) , where k ∈ { 0 , 1 } k∈ \{0,1\} , s ∈ C s∈ C , ℜ s > 0 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 ∑ n≤ XΛ (n)/n≤ log X , valid for any X ≥ 1 X≥ 1 .