Interlacing for zeros of the Serre derivative of Eisenstein series

In 1970, Rankin and Swinnerton-Dyer showed that the non-elliptic zeros of Eisenstein series $E_k$ in the fundamental domain all lie on the lower arc $\{ e^{iθ}: \fracπ{2} < θ< \frac{2π}{3}\}$. Very recently, Sugibayashi showed that the same property also holds for the Serre derivative $\vartheta_k(E_k)$ of Eisenstein series. In this paper, we first give very precise estimates for where exactly these zeros are located on the lower arc. These location estimates then allow us to prove four main results. First, we show that the zeros of $\vartheta_\ell(E_\ell)$ Stieltjes interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc for all $\ell > k$. Second, we classify precisely when the zeros of $\vartheta_\ell(E_\ell)$ (standard) interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc. Third, we show that the zeros of $\vartheta_k(E_k)$ always (standard) interlace with the zeros of $E_{k+2}$ on the lower arc. Fourth, as an application of the third main result, we show that the zeros of the cuspidal projection of $\vartheta_k(E_k)$ all lie on the lower arc, extending a result of Xue and Zhu.