Computing the Ramanujan tau function

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\frac{1}{2}+\varepsilon}$$\end{document} for every O(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\frac{3}{4}+\varepsilon}$$\end{document}) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\frac{3}{4}+\varepsilon}$$\end{document}) (without any assumptions) for every ε > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time.