Computation of the Totient Summatory Function

TL;DR

This paper introduces an efficient algorithm to compute the sum of Euler's totient function up to n, significantly improving computational speed for large n, demonstrated by calculating ^{19}.

Contribution

It presents a novel algorithm with sublinear time complexity for summing totient values, based on the Dirichlet hyperbola method and Mertens function.

Findings

Algorithm computes ^{19} in feasible time

Time complexity is approximately ^{2/3}

Space complexity is approximately ^{1/3}

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 $\Phi(10^{19}) = 30396355092701331435065976498046398788$.