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Summatory function of the Möbius function. II: Elementary asymptotic upper bounds. (Fonction sommatoire de la fonction de Möbius. II: Majorations asymptotiques élémentaires.) (French) Zbl 0817.11062

With the help of the upper bound on \(| M(x)|/ \sqrt{x}\) obtained in part I of this paper (see the previous review (Zbl 0817.11061)) it is shown that \(| M(x)| \leq {1\over 2360} x\) for \(x\geq 617973\). The proof follows N. Costa Pereira’s method [Acta Arith. 52, 307-337 (1989; Zbl 0696.10007)] which resulted in the bound \(| M(x)| \leq{1\over 1036} x\) for every \(x\geq 120727\).
The best known asymptotic bound is: \(| M(x)| <5.3 x (\log x)^{- 10/9}\) for all \(x>1\), but this is not the best bound for “intermediate” values of \(x\): for \(x<5.27\times 10^{2114}\), the new bound is better.
Remarks. The result of Lioen and van de Lune mentioned in the previous review can be used to further improve upon Dress and El Marraki’s bound.

MSC:

11Y35 Analytic computations
11Y16 Number-theoretic algorithms; complexity
11N05 Distribution of primes
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References:

[1] Cohen, H. and Dress, F. 1988.”Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteur carré”73–76. Marseille: Publications mathématiques d’Orsay: Colloque de théorie analytique des nombres. [Cohen et Dress 1988], 1985
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