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

We present a new elementary algorithm that takes $$\mathrm{time} \ \ O_\epsilon\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+\epsilon} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right)$$ for computing $M(x) = \sum_{n \leq x} \mu(n),$ where $\mu(n)$ is the M\"{o}bius 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 $x^{3/5} (\log x)$.