Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

Committee Chair/Advisor

Hui Xue

Committee Member

Keri Sather-Wagstaff

Committee Member

Michael Burr

Committee Member

Jim Coykendall


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 $e^{\pi i/3}$.

Author ORCID Identifier


Included in

Number Theory Commons



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