Explicit estimates for summatory functions linked to the Möbius $\mu$-function

Henri Cohen; Francois Dress; Mahomed El Marraki

doi:10.7169/facm/1229618741

January 2007 Explicit estimates for summatory functions linked to the Möbius $\mu$-function

Henri Cohen, Francois Dress, Mahomed El Marraki

Funct. Approx. Comment. Math. 37(1): 51-63 (January 2007). DOI: 10.7169/facm/1229618741

Abstract

Let $M(x)$ be the summatory function of the Möbius function and $R(x)$ be the remainder term for the number of squarefree integers up to $x$. In this paper, we prove the explicit bounds $|M(x)|<x/4345$ for $x\ge 2160535$ and $|R(x)|\le 0.02767\sqrt x$ for $x\ge 438653$. These bounds are considerably better than preceding bounds of the same type and can be used to improve Schoenfeld type estimates.

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Henri Cohen. Francois Dress. Mahomed El Marraki. "Explicit estimates for summatory functions linked to the Möbius $\mu$-function." Funct. Approx. Comment. Math. 37 (1) 51 - 63, January 2007. https://doi.org/10.7169/facm/1229618741