Explicit estimates on several summatory functions involving the Moebius function

Abstract:

We prove that $|\sum _{d\le x}\mu (d)/d|\log x\le 1/69$ when $x\ge 96 955$ and deduce from that: \[ \bigg |\textstyle {\sum _{\left \{\substack {d\le x,\\(d,q)=1}\right .}}\mu (d)/d\bigg |\log (x/q)\le \tfrac 45 q/\varphi (q)\] for every $x>q\ge 1$. We also give better constants when $x/q$ is larger. Furthermore we prove that $|1-\sum _{d\le x}\mu (d)\log (x/d)/d|\le \tfrac 3{14}/\log x$ 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 $\sum _{d\le x}\mu (d)\log ^2(x/d)/d$, among which we mention the bound $|\sum _{d\le x}\mu (d)\log ^2(x/d)/d-2\log x+2\gamma _0|\le \tfrac {5}{24}/\log x$, where $\gamma _0$ is the Euler constant. We complete this collection by bounds such as \[ \textstyle {\bigg |\sum _{\left \{\substack {d\le x,\\(d,q)=1}\right .}}\mu (d)\bigg |/x\le \tfrac {q}{\varphi (q)}/\log (x/q).\] We also provide all these bounds with variations where $1/\log x$ is replaced by $1/(1+\log x)$.