Computing the Summation of the Möbius Function

Abstract

We describe an elementary method for computing isolated values of\n$M(x)=\\sum_{n \\leq x} \\mu(n)$, where $\\mu$ is the Möbius function.\nThe complexity of the algorithm is $O(x^{2/3}(\\log \\log x)^{1/3})$\ntime and $O(x^{1/3}(\\log \\log x)^{2/3})$ space. Certain values of\n$M(x)$ for $x$ up to $10^{16}$ are listed: for instance, $M(10^{16})=-3195437$.