Functiones et Approximatio Commentarii Mathematici

Some elementary explicit bounds for two mollifications of the Moebius function

Olivier Ramaré

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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

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

First available in Project Euclid: 20 December 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

explicit estimates Möbius function


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.

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