Asymptotics for the Fourier coefficients of eta-quotients

TL;DR

This paper derives asymptotic formulas for the Fourier coefficients of a broad class of eta-quotients, which are products involving powers of q-series with integer exponents, expanding understanding of their growth behavior.

Contribution

The paper provides new asymptotic formulas for Fourier coefficients of eta-quotients with general parameters, extending previous results to a wider class of functions.

Findings

Derived explicit asymptotic formulas for Fourier coefficients

Extended known results to eta-quotients with multiple parameters

Enhanced understanding of growth rates of eta-quotient coefficients

Abstract

We study the asymptotics for the Fourier coefficients of a broad class of eta-quotients, $$\prod_{r=1}^R \left(\prod_{k\ge 1}\left(1-q^{m_r k}\right)\right)^{\delta_r},$$ where $m_1,\ldots,m_R$ are $R$ distinct positive integers and $\delta_1,\ldots,\delta_R$ are $R$ non-zero integers with $\sum_{r=1}^R \delta_{r}\ge 0$.