A modified Dirichlet-hyperbola algorithm computes Φ(n) in Θ(n^{2/3}(ln ln n)^{1/3}) time and Θ(n^{1/3}(ln ln n)^{2/3}) space, and produced the first computation of Φ(10^19).
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Computation of the Totient Summatory Function
A modified Dirichlet-hyperbola algorithm computes Φ(n) in Θ(n^{2/3}(ln ln n)^{1/3}) time and Θ(n^{1/3}(ln ln n)^{2/3}) space, and produced the first computation of Φ(10^19).