## 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

### Citations:

Zbl 0176.32502; Zbl 0817.11062
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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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