# The ACO Seminar (2018–2019)

March 8, 2019, 3:30pm, Wean 8220
Harald Helfgott, University of Gottingen
Summing $\mu(n)$: a better elementary algorithm

Abstract:

Consider either of two related problems: determining the precise number $\pi(x)$ of prime numbers $p\leq x$, and computing the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$, where $\mu$ is the M\"obius function. The two best algorithms known are the following: * An analytic algorithm (Lagarias-Odlyzko, 1987), with computations based on integrals of $\zeta(s)$; its running time is $O(x^{1/2+\epsilon})$. * A more elementary algorithm (Meissel-Lehmer, 1959 and Lagarias-Miller-Odlyzko, 1985; refined by Del\'eglise-Rivat, 1996), with running time about $O(x^{2/3})$. The analytic algorithm had to wait for almost 30 years to receive its first rigorous, unconditional implementation (Platt), which concerns only the computation of $\pi(x)$. Moreover, in the range explored to date ($x\leq 10^{24}$), the elementary algorithm is faster in practice. We present a new elementary algorithm with running time about $O(x^{3/5})$ for computing $M(x) = \sum_{n\leq x} \mu(n)$. The algorithm should be adaptable to computing $\pi(x)$ and other related problems.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.