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Explicit estimates on several summatory functions involving the Moebius function


Author: Olivier Ramaré
Journal: Math. Comp. 84 (2015), 1359-1387
MSC (2010): Primary 11N37, 11Y35; Secondary 11A25
Published electronically: December 1, 2014
MathSciNet review: 3315512
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Abstract: We prove that $ \vert\sum _{d\le x}\mu (d)/d\vert\log x\le 1/69$ when $ x\ge 96\,955$ and deduce from that:

$\displaystyle \bigg \vert\textstyle {\sum _{\left \{\substack {d\le x,\\ (d,q)=1}\right .}}\mu (d)/d\bigg \vert\log (x/q)\le \tfrac 45 q/\varphi (q)$

for every $ x>q\ge 1$. We also give better constants when $ x/q$ is larger. Furthermore we prove that $ \vert 1-\sum _{d\le x}\mu (d)\log (x/d)/d\vert\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 $ \vert\sum _{d\le x}\mu (d)\log ^2(x/d)/d-2\log x+2\gamma _0\vert\le \tfrac {5}{24}/\log x$, where $ \gamma _0$ is the Euler constant. We complete this collection by bounds such as

$\displaystyle \textstyle {\bigg \vert\sum _{\left \{\substack {d\le x,\\ (d,q)=1}\right .}}\mu (d)\bigg \vert/x\le \tfrac {q}{\varphi (q)}/\log (x/q).$

We also provide all these bounds with variations where $ 1/\log x$ is replaced by $ 1/(1+\log x)$.

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Additional Information

Olivier Ramaré
Affiliation: Laboratoire CNRS Paul Painlevé, Université Lille 1, 59655 Villeneuve d’Ascq, France
Email: ramare@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0025-5718-2014-02914-1
Keywords: Explicit estimates, Moebius function
Received by editor(s): March 27, 2013
Received by editor(s) in revised form: November 9, 2013
Published electronically: December 1, 2014
Article copyright: © Copyright 2014 American Mathematical Society