A note on zeros of Eisenstein series for genus zero Fuchsian groups

Abstract

Let $\Gamma \subseteq \text{SL}_2(\mathbb{R})$ be a genus zero Fuchsian group of the first kind having $\infty$ as a cusp, and let $E_{2 k}^{\Gamma}$ be the holomorphic Eisenstein series associated with $\Gamma$ for the $\infty$ cusp that does not vanish at $\infty$ but vanishes at all the other cusps. In the paper "On zeros of Eisenstein series for genus zero Fuchsian groups", under assumptions on $\Gamma$, and on a certain fundamental domain $\mathcal{F}$, H. Hahn proved that all but at most $c(\Gamma, \mathcal{F})$ (a constant) of the zeros of $E_{2 k}^{\Gamma}$ lie on a certain subset of $\{z \in \mathfrak{H} : j_{\Gamma}(z) \in \mathbb{R}\}$. In this note, we consider a small generalization of Hahn's result on the domain locating the zeros of $E_{2 k}^{\Gamma}$. We can prove most of the zeros of $E_{2 k}^{\Gamma}$ in $\mathcal{F}$ lie on its lower arcs under the same assumption.