艾森斯坦级数的塞尔导数零点的交错性
Interlacing for zeros of the Serre derivative of Eisenstein series
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中文总结 AI 辅助
本文研究艾森斯坦级数的塞尔导数零点的交错性，通过精确估计零点位置，证明了其零点与其他相关级数零点的Stieltjes交错等四个结果，扩展了Xue和Zhu的相关结论。
中文摘要 AI 辅助
1970年，Rankin和Swinnerton-Dyer证明，艾森斯坦级数$E_k$在基本域中的非椭圆零点全部位于下弧$\{e^{i\theta}: \frac{\pi}{2} < \theta < \frac{2}{3}\pi\}$上。最近，Sugibayashi证明该性质同样适用于艾森斯坦级数的塞尔导数$\u001f_k(E_k)$。本文首先对这些零点在下弧上的精确位置给出非常精确的估计，这些位置估计使我们能够证明四个主要结果：第一，对所有$\u001e > k$，$\u001f_\u001e(E_\u001e)$的零点与$\u001f_k(E_k)$的零点在下弧上是Stieltjes交错的；第二，我们精确分类$\u001f_\u001e(E_\u001e)$的零点何时与$\u001f_k(E_k)$的零点在下弧上是（标准）交错的；第三，我们证明$\u001f_k(E_k)$的零点总是与$E_{k+2}$的零点在下弧上是（标准）交错的；第四，作为第三个主要结果的应用，我们证明$\u001f_k(E_k)$的尖点投影的零点全部位于下弧上，扩展了Xue和Zhu的结果。
英文摘要
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.
