Asymptotic formulas for the coefficients of certain automorphic functions

Jaban Meher¹ ; Karam Deo Shankhadhar²

¹ Department of Mathematics Indian Institute of Science Bangalore 560012, India
² Departamento de Matemáticas Facultad de Ciencias Universidad de Chile Las Palmeras 3425 Nunoa, Santiago, Chile

Acta Arithmetica, Tome 169 (2015) no. 1, pp. 59-76.

Résumé

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.

Détail

DOI : 10.4064/aa169-1-4

Keywords: derive asymptotic formulas coefficients certain classes weakly holomorphic jacobi forms weakly holomorphic modular forms necessarily integral weight without using circle method applications these formulas given namely estimate growth fourier coefficients important weak jacobi forms index non positive weights obtain asymptotic formula fourier coefficients modular functions theta eta integers geq where theta nbsp weight nbsp modular form eta nbsp dedekind eta function