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arXiv · 2101.08773

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

Harald A. Helfgott·Lola Thompson
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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 $x^{3/5} (\log x)$.

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Harald A. Helfgott, Lola Thompson. 2021-01-21. Summing $\mu(n)$: a faster elementary algorithm. https://arxiv.org/abs/2101.08773

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