Computing the Ramanujan tau function

Abstract

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time \(n^{\frac{1}{2}+\varepsilon}\) for every O(\(n^{\frac{3}{4}+\varepsilon}\)) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O(\(n^{\frac{3}{4}+\varepsilon}\)) (without any assumptions) for every e > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time.