On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms

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 $|\tau(n)|\le n^{11/2} (\log n)^{-1/2+o(1)}$ 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. 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.