On the zeros of a class of modular functions

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy $$f_k(A,\tau )\text{\hspace{0.17em}⁣}:=q^{-k}(1+a(1)q+a(2)q^2+\mathrm{.}\mathrm{.}\mathrm{.})+O(q),$$ where $a(n)$ are numbers satisfying a certain analytic condition. We show that the zeros of such $f_k(\tau )$ in the fundamental domain of $S{L}_2(\mathbb{Z})$ lie on $\mathrm{\mid }\tau \mathrm{\mid }=1$ and are transcendental. We recover as a special case earlier work of Witten on extremal “partition” functions $Z_k(\tau )$. These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity.