# Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l

## TL;DR

This paper determines the location of zeros of specific combinations of Eisenstein series of weights 2k, 3k, and k+l, showing that for large weights, all zeros lie on a particular boundary segment of the fundamental domain.

## Contribution

It extends previous work by locating zeros of certain Eisenstein series combinations, proving they lie on a specific boundary for large weights.

## Key findings

- Zeros are located on the boundary segment for large weights.
- All zeros in the fundamental domain are on the lower boundary for sufficiently large k,l.
- The results build on Rankin and Swinnerton-Dyer's work.

## Abstract

We locate the zeros of the modular forms $E_k^2(\tau) + E_{2k}(\tau), E_k^3(\tau) + E_{3k} (\tau),$ and $E_k(\tau)E_l(\tau) +E_{k+l}(\tau),$ where $E_k(\tau)$ is the Eisenstein series for the full modular group $\text{SL}_2(\mathbb{Z})$. By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, we prove that for sufficiently large $k,l$, all zeros in the standard fundamental domain are located on the lower boundary $\mathcal{A} = \{ e^{i\theta} : \pi/2 \leq \theta \leq 2\pi/3\}$.