Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l
We locate the zeros of the modular forms $E_k^2(τ) + E_{2k}(τ), E_k^3(τ) + E_{3k} (τ),$ and $E_k(τ)E_l(τ) +E_{k+l}(τ),$ where $E_k(τ)$ 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 $\mathcal{A} = \{ e^{iθ} : π/2 \leq θ\leq 2π/3\}$.
math.NT↗