Some elementary explicit bounds for two mollifications of the Moebius function

Some elementary explicit bounds for two mollifications of the Moebius function O. Ramare March 8, 2012 Abstract We prove that the sum ∑ { d≤x, (d,r)=1 µ(d)/d1+? is bounded by 1 + ?, uniformly in x ≥ 1, r and ? > 0. We prove a similar estimate for the quantity ∑ { d≤x, (d,r)=1 µ(d) Log(x/d)/d1+?. When ? = 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. 1 Introduction and results Our first result is the following: Theorem 1.1. When r ≥ 1 and ? ≥ 0, we have ? ? ? ? ∑ d≤x, (d,r)=1 µ(d) d1+? ? ? ? ? ≤ 1 + ?. This Lemma generalizes the estimate of [5, Lemme 10.2] which corre- sponds to the case ? = 0. This generalization is not straightforward at all and requires a change of proof. The case ? = 0 and r = 1 is classical.

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