We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight \(\frac{g}{2}\) and genus g. When \(g=4\), the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.

中文翻译：

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权重 2 的 Eisenstein 级数的导数和模对应的交集

我们给出了 Siegel 级数的某些值和导数的公式，并使用它们来计算 Siegel Eisenstein 级数的权重\(\frac{g}{2}\)和属g的导数的傅立叶系数。当\(g=4\)时，傅立叶系数近似为权重为 2 和属 3 的 Siegel Eisenstein 级数的中心导数的某个傅立叶系数，这与 3 个算术模对应的交集有关。应用包括对称矩阵的表示数的加权平均值之间的关系。