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)∣≤n11/2(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 log2 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 assuming the automorphy of all symmetric powers of the form and
seems to be within reach unconditionally using the currently established
potential automorphy.Comment: The second version contains some improvements and extensions of
previous results, suggested by Maksym Radziwill, who is now a co-autho