Summing $\mu (n)$: a faster elementary algorithm

Abstract

We present a new elementary algorithm that takes \[ \mathrm{time} \ \ O_ε\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+ε} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right)\] for computing $M(x)={\sum }_{n\le x}\mu (n),$ where $\mu (n)$ is the Möbius function. This is the first improvement in the exponent of $x$ for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $O(x^{1/5}(\log x{)}^{5/3})$ by the use of (Helfgott, 2020; arxiv.org:1712.09130), at the cost of letting time rise to the order of $x^{3/5}(\log x)$.

43 pages, 2 figures. Fourth version: streamlined abstract, rewritten introduction