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

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.