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

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 $\mathcal{A}=\{e^{i\theta }:\pi /2\le \theta \le 2\pi /3\}$.

References in corpus (1)

• On the zeros of Eisenstein series for ${\Gamma }_0^{\ast }(2)$ and ${\Gamma }_0^{\ast }(3)$