Computation of the Totient Summatory Function

Lucas Augustus Brown

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arXiv · 2506.07386

Computation of the Totient Summatory Function

Lucas Augustus Brown

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

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Lucas Augustus Brown. 2025-06-09. Computation of the Totient Summatory Function. https://arxiv.org/abs/2506.07386

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