## Abstract

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ≤ n ^{11/2} (logn) ^{-1/2+o(1)} for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of -1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n. In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato-Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

Original language | English (US) |
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Pages (from-to) | 173-189 |

Number of pages | 17 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 166 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

## ASJC Scopus subject areas

- General Mathematics