From explicit estimates for the primes to explicit estimates for the Moebius function

From explicit estimates for the primes to explicit estimates for the Moebius function O. Ramare March 8, 2012 Abstract We prove an estimate slightly stronger than | ∑ d≤D µ(d)/d| ≤ 0.03/LogD for every D ≥ 11 815. 1 Introduction There is a long litterature concerning explicit estimates for the summatory function of the Moebius function, and we cite for instance [20], [1], [4], [3], [6], [7], [10], [11]. The paper [5] proposes a very usefull annoted bibliography covering relevant items up to 1983. It has been known since the beginning of the 20th century at least (see for instance [13]) that showing that M(x) = ∑ n≤x µ(n) is o(x) is equivalent to showing that the Tchebychef function ?(x) = ∑ n≤x ?(n) is asymptotic to x. We have good explicit estimates for ?(x) ? x, see for instance [18], [21] and [9]. This is due to the fact that we can use analytic tools in this problem since the residues at the poles of the Dirichlet generating series (namely here ?? ?(s)/?(s)) are known.

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