Some elementary explicit bounds for two mollifications of the Moebius function
Abstract
We prove that the $\sum_{d\le x, (d,r)=1}\mu(d)/d^{1+\varepsilon}$ is bounded by $1 + \varepsilon$, uniformly in $x \ge 1$, $r$ and $\varepsilon > 0$. We prove a similar estimate for the quantity $\sum_{d\le x, (d,r)=1} \mu(d) \log(x/d)/d^{1+\varepsilon}. When $\varepsilon=0$, $r$ 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.
Domains
Number Theory [math.NT]
Fichier principal
MuLog-4.pdf (271.93 Ko)
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MuLog-4 (1).pdf (271.93 Ko)
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