Functiones et Approximatio Commentarii Mathematici

Some elementary explicit bounds for two mollifications of the Moebius function

Olivier Ramaré

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Abstract

We prove that the sum $\sum_{\left\{d\le x, (d,r)=1\right.}\mu(d)/d^{1+\varepsilon}$ is bounded by $1+\varepsilon$, uniformly in $x\ge1$, $r$ and $\varepsilon>0$. We prove a similar estimate for the quantity $\sum_{\left\{d\le x, (d,r)=1\right.}\mu(d)\log(x/d)/d^{1+\varepsilon}$. When $\varepsilon=0$, $r$ varies between 1 and a hundred, and $x$ is below a million, this sum is non-negative and this raises the question as to whether it is non-negative for every~$x$.

Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 2 (2013), 229-240.

Dates
First available in Project Euclid: 20 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1387572228

Digital Object Identifier
doi:10.7169/facm/2013.49.2.3

Mathematical Reviews number (MathSciNet)
MR3161492

Zentralblatt MATH identifier
1288.11091

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions 11Y35: Analytic computations
Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas

Keywords
explicit estimates Möbius function

Citation

Ramaré, Olivier. Some elementary explicit bounds for two mollifications of the Moebius function. Funct. Approx. Comment. Math. 49 (2013), no. 2, 229--240. doi:10.7169/facm/2013.49.2.3. https://projecteuclid.org/euclid.facm/1387572228


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