Summing μ ( n ) : a faster elementary algorithm.

Abstract

We present a new elementary algorithm that takes $\text{time}{O}_{ϵ}\left(x^{\frac{3}{5}},{(logx)}^{\frac{8}{5}+ϵ}\right)\text{and}\text{space}O\left(x^{\frac{3}{10}},{(logx)}^{\frac{13}{10}}\right)$ (measured bitwise) for computing $M\left(x\right)={\sum }_{n\le x}\mu \left(n\right),$ where $\mu \left(n\right)$ 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 $O\left(x^{1/5}{(logx)}^{5/3}\right)$ by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of $x^{3/5}{(logx)}^{2}loglogx$ .