Computation of the Totient Summatory Function

Abstract

An algorithm is devised for computing $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$ in time $\widetilde{\Theta}(n^{2/3})$ and space $\widetilde{\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet hyperbola method and the Mertens function. The algorithm is then used to compute $\Phi(10^{19}) = 30396355092701331435065976498046398788$.