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

TL;DR
This paper extends previous results on the zeros of Eisenstein series for genus zero Fuchsian groups, showing most zeros lie on the lower arcs of a fundamental domain under certain conditions.
Contribution
It generalizes Hahn's results by demonstrating that most zeros of Eisenstein series are located on the lower arcs of the fundamental domain.
Findings
Most zeros of $E_{2k}^ ext{Gamma}$ lie on the lower arcs of the fundamental domain.
The result holds under similar assumptions as Hahn's original theorem.
The zeros are confined to a specific subset related to the real axis of the $j_ ext{Gamma}$-invariant.
Abstract
Let be a genus zero Fuchsian group of the first kind having as a cusp, and let be the holomorphic Eisenstein series associated with 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 , and on a certain fundamental domain , H. Hahn proved that all but at most (a constant) of the zeros of lie on a certain subset of . In this note, we consider a small generalization of Hahn's result on the domain locating the zeros of . We can prove most of the zeros of in lie on its lower arcs under the same assumption.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Advanced Algebra and Geometry
