Computing the summation of the Möbius function

Abstract

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