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Examples of such sequences come from coefficients of several $L$-functions of elliptic curves and modular forms. In particular, we show that $|\\tau(n)|\\le n^{11/2} (\\log n)^{-1/2+o(1)}$ for a set of $n$ of asymptotic density 1, where $\\tau(n)$ is the Ramanujan $\\tau$ function while the standard argument yields $\\log 2$ instead of $-1/2$ in the power of the logarithm. 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