A Note on Zeros of Eisenstein Series for Genus Zero Fuchsian Groups

Let Γ ⊆ SL2(R) be a genus zero Fuchsian group of the first kind having ∞ as a cusp, and let EΓ 2k be the holomorphic Eisenstein series associated with Γ for the ∞ cusp that does not vanish at ∞ but vanishes at all the other cusps. In the paper “On zeros of Eisenstein series for genus zero Fuchsian groups”, under assumptions on Γ, and on a certain fundamental domain F , H. Hahn proved that all but at most c(Γ,F) (a constant) of the zeros of EΓ 2k lie on a certain subset of {z ∈ H : jΓ(z) ∈ R}. In this note, we consider a small generalization of Hahn’s result on the domain locating the zeros of EΓ 2k. We can prove most of the zeros of E Γ 2k in F lie on its lower arcs under the same assumption.