Computing the Mertens function on a GPU

Abstract

A GPU implementation of an algorithm to compute the Mertens function in O(x2/3+{\ko}) time is discussed. Results for x up to $10^{22}$, and a new extreme value for $M(x)/x^{1/2}$, -0.585768 ($M(x)\approx -1.996\ast 10^9$ at $x\approx 1.161\ast 10^{19}$), are reported.An approximate algorithm is used to examine values of M(x) for x up to $\exp (10^{15})$.