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