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

TL;DR

This paper establishes upper bounds and cancellation properties for sequences of coefficients from modular forms and related L-functions, demonstrating typical size constraints and a central limit theorem under certain distribution assumptions.

Contribution

It provides new upper bounds for the coefficients of modular forms and L-functions, showing typical size and cancellation properties for almost all integers, with results applicable to Ramanujan tau and quadratic form representations.

Findings

Upper bound for Ramanujan tau function coefficients for almost all n

Demonstration of more than square-root cancellations in quadratic form representations

A central limit theorem for these sequences under a weak distribution hypothesis

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