# 艾森斯坦級數

## 模群的艾森斯坦級數

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

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

## 遞迴關係

$d_k := (2k+3)k!G_{2k+4}$，遂有下述關係式：

$\sum_{k=0}^n {n \choose k} d_k d_{n-k} = \frac{2n+9}{3n+6}d_{n+2}$

$\wp(z) =\frac{1}{z^2} + z^2 \sum_{k=0}^\infty \frac {d_k z^{2k}}{k!} =\frac{1}{z^2} + \sum_{k=1}^\infty (2k+1) G_{2k+2} z^{2k}$

## 傅立葉展開

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

$G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)$

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

$G_4(\tau)=\frac{\pi^4}{45} \left[ 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n} \right]$
$G_6(\tau)=\frac{2\pi^6}{945} \left[ 1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n} \right]$

$|q|<1$，對 $q$ 之和亦可化成蘭伯特級數

$\sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n}$

$E_{2k} := \frac{G_{2k}}{2 \zeta(2k)} = 1 - \frac{4k}{B_{2k}} \sum_{n=1}^\infty \sigma_{2k-1}(n) q^n$

### 拉馬努金公式

$L(q)=1-24\sum_{n=1}^\infty \frac {nq^n}{1-q^n} = E_2(\tau)$
$M(q)=1+240\sum_{n=1}^\infty \frac {n^3q^n}{1-q^n} = E_4(\tau)$
$N(q)=1-504\sum_{n=1}^\infty \frac {n^5q^n}{1-q^n} = E_6(\tau)$

$q\frac{dL}{dq} = \frac {L^2-M}{12}$
$q\frac{dM}{dq} = \frac {LM-N}{3}$
$q\frac{dN}{dq} = \frac {LN-M^2}{2}$

## 文獻

• 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.