On the Zeroes of Half-Integral Weight Eisenstein Series on ${\Gamma }_0(4)$

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.

13 pages, 6 images (3 figures)