Some elementary explicit bounds for two mollifications of the Moebius function

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~$x$.