艾森斯坦級數

固定整數 ${\displaystyle k>1}$。對上半平面上的複數 ${\displaystyle \tau }$，定義艾森斯坦級數 ${\displaystyle G_{2k}}$ 為

${\displaystyle G_{2k}(\tau )=\sum _{(m,n)\neq (0,0)}{\frac {1}{(m+n\tau )^{2k}}}.}$

此級數是上半平面上的全純函數，此外它更是模群 ${\displaystyle \Gamma :=\mathrm {SL} (2,\mathbb {Z} )}$ 的權 ${\displaystyle 2k}$ 模形式。換言之，若 ${\displaystyle a,b,c,d\in \mathbb {Z} }$ 滿足 ${\displaystyle ad-bc=1}$，則

${\displaystyle G_{2k}\left({\frac {a\tau +b}{c\tau +d}}\right)=(c\tau +d)^{2k}G_{2k}(\tau )}$

模形式理論中的一個基本事實是：模群 ${\displaystyle \Gamma }$ 的模形式俱可表為 ${\displaystyle G_{4}}$ 與 ${\displaystyle G_{6}}$ 的多項式。作為特例，以下說明如何將艾森斯坦級數遞迴地表成 ${\displaystyle G_{4},G_{6}}$ 的多項式。

置 ${\displaystyle d_{k}:=(2k+3)k!G_{2k+4}}$，遂有下述關係式：

${\displaystyle \sum _{k=0}^{n}{n \choose k}d_{k}d_{n-k}={\frac {2n+9}{3n+6}}d_{n+2}}$

在此 ${\displaystyle {n \choose k}}$ 是二項式係數而 ${\displaystyle d_{0}=3G_{4}}$、${\displaystyle d_{1}=5G_{6}}$。

函數 ${\displaystyle d_{k}}$ 可以表示魏爾斯特拉斯 ${\displaystyle \wp }$ 函數：

${\displaystyle \wp (z)={\frac {1}{z^{2}}}+z^{2}\sum _{k=0}^{\infty }{\frac {d_{k}z^{2k}}{k!}}={\frac {1}{z^{2}}}+\sum _{k=1}^{\infty }(2k+1)G_{2k+2}z^{2k}}$

置 ${\displaystyle q=e^{2\pi i\tau }}$。由於艾森斯坦級數是模群的模形式，故有傅立葉展開式

${\displaystyle G_{2k}(\tau )=2\zeta (2k)\left(1+c_{2k}\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}\right)}$

其中的傅立葉係數 ${\displaystyle c_{2k}}$ 是

${\displaystyle c_{2k}={\frac {(2\pi i)^{2k}}{(2k-1)!\zeta (2k)}}={\frac {-4k}{B_{2k}}}}$。

此處的 ${\displaystyle B_{n}}$ 是伯努利數，${\displaystyle \zeta (z)}$是黎曼ζ函數，而 ${\displaystyle \sigma _{p}(n)}$ 是 ${\displaystyle n}$ 的正因數的 ${\displaystyle p}$ 次冪和。

${\displaystyle G_{4}(\tau )={\frac {\pi ^{4}}{45}}\left[1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right]}$

${\displaystyle G_{6}(\tau )={\frac {2\pi ^{6}}{945}}\left[1-504\sum _{n=1}^{\infty }\sigma _{5}(n)q^{n}\right]}$

當 ${\displaystyle |q|<1}$，對 ${\displaystyle q}$ 之和亦可化成蘭伯特級數

${\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}}$。

有時也會考慮常數項等於一的艾森斯坦級數：

${\displaystyle E_{2k}:={\frac {G_{2k}}{2\zeta (2k)}}=1-{\frac {4k}{B_{2k}}}\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$。

• Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0
• Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 (See chapter 3)
• Jean-Pierre Serre, A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.