Explicit estimates on several summatory functions involving the Moebius function

We prove that | ∑ d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that: ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d)/d ∣ ∣ ∣ ∣ log(x/q) ≤ 4 5 q/φ(q) for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − ∑ d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with the additional condition (d, q) = 1. We prove similar results for ∑ d≤x μ(d) log (x/d)/d, among which we mention the bound | ∑ d≤x μ(d) log (x/d)/d − 2 log x + 2γ0| ≤ 5 24/ log x, where γ0 is the Euler constant. We complete this collection by bounds such as ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d) ∣ ∣ ∣ ∣/x ≤ q φ(q) / log(x/q). We also provide all these bounds with variations where 1/ log x is replaced by 1/(1 + log x).