normalized full-level holomorphic Eisenstein series

全模群的正规化艾森斯坦级数

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Other names / variants: normalized Eisenstein series

Translation evidence: Literature usage

Meaning in context

For even $k\ge4$ and $\operatorname{Im}\tau>0$, $E_k(\tau)=\tfrac12\sum_{(c,d)\in\mathbb Z^2,\ \gcd(c,d)=1}(c\tau+d)^{-k}$, an absolutely convergent weight-$k$ modular form with constant term $1$.

对偶数 $k\ge4$ 及 $\operatorname{Im}\tau>0$，$E_k(\tau)=\tfrac12\sum_{(c,d)\in\mathbb Z^2,\ \gcd(c,d)=1}(c\tau+d)^{-k}$ 是绝对收敛、常数项为 $1$ 的权 $k$ 模形式。

Example

$E_4(\tau)=1+240q+2160q^2+\cdots$, $q=e^{2\pi i\tau}$.

Usage and distinctions

This normalization divides the unrestricted lattice sum by 2 zeta(k); weight 2 needs a different treatment and is not included in the convergent definition.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 4.3.3, Definition 4.37 and Proposition 4.38, printed page 117; k restricted to at least 4 · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义定义4.37 艾森斯坦级数，命题4.38，printed page 117 · Checked 2026-10-14

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