On the Zeroes of Half-Integral Weight Eisenstein Series on $\Gamma_0(4)$

TL;DR

This paper analyzes the distribution of zeros of half-integral weight Eisenstein series on (4), providing an approximate location of all but a small number of zeros for large weights by relating different Eisenstein series and using trigonometric approximations.

Contribution

It introduces a method to locate zeros of half-integral weight Eisenstein series on (4) for large weights, improving understanding of their zero distribution.

Findings

Locates all but O((k   k)) zeros of the Eisenstein series for large k

Relates different Eisenstein series to facilitate zero location

Uses trigonometric approximations to identify zero positions

Abstract

We locate all but $O(\sqrt{k\log{k}})$ zeroes of the half integral weight Eisenstein series $E_\infty(z,k)$ of $\Gamma_0(4)$ for $k$ sufficiently large. To do this, we relate $E_\infty(z,k)$ to $\Gamma_0(4)$'s other Eisenstein series, $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$, which are easier to study in the regions of which zeroes occur. We will use trigonometric approximations of $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$ in order to locate the zeroes.