Date of Award
8-2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
Hui Xue
Committee Member
Keri Sather-Wagstaff
Committee Member
Michael Burr
Committee Member
Jim Coykendall
Abstract
Let $E_k(z)$ be the normalized Eisenstein series of weight $k$ for the full modular group $\text{SL}(2, \mathbb{Z})$. It is conjectured that all the zeros of the weight $k+\ell$ cusp form $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. Reitzes, Vulakh and Young \cite{Reitzes17} proved that this statement is true for sufficiently large $k$ and $\ell$. Xue and Zhu \cite{Xue} proved the cases when $\ell=4,6,8$ with $k\geq\ell$, all the zeros of $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ lie on the arc $|z|=1$. For all $k\geq\ell\geq10$, we will use the same method as \cite{Reitzes17} to locate these zeros on the arc $|z|=1$, and improve the method in \cite{Reitzes17} to locate these zeros on the side boundary $\{\frac{1}{2}+iy : y>\frac{\sqrt{3}}{2}\}$. At last, we will use the method in \cite[Theorem 1.1]{DoubleInterlace} to find the extra zeros close to $e^{\pi i/3}$.
Recommended Citation
Zhu, Daozhou, "Zeros of Modular Forms" (2023). All Dissertations. 3385.
https://tigerprints.clemson.edu/all_dissertations/3385
Author ORCID Identifier
0000-0002-9089-1091