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Summatory function of the Möbius function. III: Strong effective asymptotic upper bounds. (Fonction sommatoire de la fonction de Möbius. III: Majorations asymptotiques effectives fortes.) (French) Zbl 0869.11075

Let \(\mu(x)\) be the well-known Möbius function, and \(M(x)=\sum_{1\leq n\leq x}\mu(n)\). In this paper asymptotic upper bounds on \(|M(x)|\) of the form \(|M(x)|<c_\alpha x/(\log x)^\alpha\) are proved for a variety of rational values of \(\alpha\in(0,236/75]\). For example, \[ \begin{alignedat}{2} |M(x)|&\leq0.002969 x/(\log x)^{1/2}&&\quad \text{for} x\geq 142194,\\ |M(x)|&\leq 0.6437752 x/\log x &&\quad \text{for} x>1\quad \text{and}\\ |M(x)|&\leq 12590292 x/(\log x)^{236/75} &&\quad \text{for} x>1. \end{alignedat} \] These bounds improve similar ones of L. Schoenfeld [Acta Arith. 15, 221-233 (1969; Zbl 0176.32502)].
In addition it is shown how effective upper bounds of the form \(|M(x)|\leq c_kx(\log\log x)^{2k}/(\log x)^k\) where \(k\) is a positive integer, can be derived. This is effectuated in examples like \(|M(x)|\leq 0.0149 x(\log\log x)^2/\log x\) for \(x\geq 2855\) and \(|M(x)|\leq 2.3132 x(\log\log x)^4/(\log x)^2\) for \(x\geq 10\).
The results in this paper are based on an improved method of Schoenfeld from the paper mentioned above, and on results of F. Dress and M. El Marraki [Fonction sommatoire de la fonction de Möbius. II: Majorations asymptotiques élémentaires, Exp. Math. 2, No. 2, 99-112 (1993; Zbl 0817.11062)] and of H. Cohen, F. Dress and M. El Marraki [Explicit estimates for summatory functions linked to the Möbius \(\mu\)-function, submitted for publication].

MSC:

11N37 Asymptotic results on arithmetic functions
11Y35 Analytic computations
11N05 Distribution of primes
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References:

[1] Cohen, H., Dress, F. et El Marraki, M., Explicit estimates for summatory functions linked to the Môbius J.L-Function ( soumis à Maths of computation). · Zbl 1230.11118
[2] Dress, F. et El Marraki, M., Fonction sommatoire de la fonction de Möbius 2. Majorations asymptotiques élémentaires, Experimental Mathematics, 2 (1993), n° 2, p. 99-112. · Zbl 0817.11062
[3] El Marraki, M., Majorations effectives de la fonction sommatoire de la fonction de Möbius, Thèse Univ. Bordeaux (1991). · Zbl 0869.11075
[4] Möbius, A.F., Uber eine besondere Art von Untersuchrung des Reihen, J. reine Angew. Math.9 (1832), p. 105-123.
[5] Schoenfeld, L., An improved. estimate for the summatory function of the Möbius function, Acta Arithmetica15 (1960), p. 221-233. · Zbl 0176.32502
[6] Schoenfeld, L., Sharper bounds for the Chebyshev functions θ(x) and Ψ(x).II mathematics of computation, volume 30, number 134 april (1976), p. 337-360. · Zbl 0326.10037
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