Summing mu (n): a faster elementary algorithm

Abstract

We present a new elementary algorithm that takes textrm{time} O_epsilon left( x^{frac{3}{5}} (log x)^{frac{8}{5}+epsilon } right) textrm{and} textrm{space} Oleft( x^{frac{3}{10}} (log x)^{frac{13}{10}} right) (measured bitwise) 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 in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of x^{3/5} (log x)^2 log log x.