Records on the vanishing of Fourier coefficients of Powers Of the Dedekind Eta Function

TL;DR

This paper extends known results on the vanishing and non-vanishing of Fourier coefficients of powers of the Dedekind eta function, addressing conjectures and providing extensive numerical evidence.

Contribution

It significantly extends Serre's table, addresses conjectures of Cohen, Strömberg, and Ono, and relates non-vanishing of coefficients to Maeda's conjecture.

Findings

All Fourier coefficients a_9(n) are non-zero for n ≤ 10^10.

Extended Serre's table on vanishing properties.

Connected non-vanishing of Δ^2 coefficients to Maeda's conjecture.

Abstract

In this paper we significantly extend Serre's table on the vanishing properties of Fourier coefficients of odd powers of the Dedekind eta function. We address several conjectures of Cohen and Str\"omberg and give a partial answer to a question of Ono. In the even-power case, we extend Lehmer's conjecture on the coefficients of the discriminant function $\Delta$ to all non-CM-forms. All our results are supported with numerical data. For example all Fourier coefficients $a_9(n)$ of the $9$-th power of the Dedekind eta function are non-vanishing for $n \leq 10^{10}$. We also relate the non-vanishing of the Fourier coefficients of $\Delta^2$ to Maeda's conjecture.