Summatory function of the Möbius function. III: Strong effective asymptotic upper bounds
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 \textit{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 \textit{F. Dress} and \textit{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 \textit{H. Cohen, F. Dress} and \textit{M. El Marraki} [Explicit estimates for summatory functions linked to the Möbius \(\mu\)-function, submitted for publication].
scientific article; zbMATH DE number 858572