On certain generalized modular forms
This paper is concerned with a characterization of certain generalized modular forms studied by \textit{M.~Knopp} and \textit{G.~Mason} [J. Number Theory 99, No. 1, 1--28 (2003; Zbl 1074.11025)]. Let \(f\) be a generalized modular form for \(\Gamma_0 (N)\) of weight zero, which has an infinite product expansion of the form \[ f(z) = q^h \prod^\infty_{n=0} (1 - q^n)^{c(n)} \] with \(q= e^{2\pi i z}\). The author proves that div\((f) \neq \emptyset\) if and only if \[ c (n) \ll_\varepsilon n^{- \frac12 +\varepsilon} \] with \(\varepsilon >0\). This result implies that the exponents \(c(n)\) take infinitely many distinct values for such a modular form with \(\mathrm{div}(f) \neq \emptyset\).
scientific article; zbMATH DE number 5798752