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

TL;DR

This paper introduces a new elementary algorithm for computing the M"obius sum $M(x)$ with improved time complexity, marking the first such improvement since 1985, and discusses space reduction techniques with trade-offs.

Contribution

It presents the first elementary algorithm since 1985 with improved exponent for computing $M(x)$, and explores space reduction methods with associated time trade-offs.

Findings

New elementary algorithm with $O_\epsilon(x^{3/5} (\log x)^{3/5+\epsilon})$ time

Space can be reduced to $O(x^{1/5} (\log x)^{5/3})$ with increased time

First improvement in exponent for elementary $M(x)$ computation since 1985

Abstract

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 $x^{3/5} (\log x)$.