Asymptotic formulas for the coefficients of certain automorphic functions
Volume 169 / 2015
Acta Arithmetica 169 (2015), 59-76
MSC: Primary 11F30; Secondary 11F50, 11F03.
DOI: 10.4064/aa169-1-4
Abstract
We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index $1$ and non-positive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions $\theta ^k/\eta ^l$ for all integers $k,l\geq 1$, where $\theta $ is the weight $1/2$ modular form and $\eta $ is the Dedekind eta function.
