在數學 中，艾森斯坦級數 是一類可直接表成級數 的模形式 ，由費迪南·艾森斯坦 首創。對於一般的約化群 ，羅拔·朗蘭茲 也發展了相應的理論。
固定整數
k
>
1
{\displaystyle k>1}
τ
{\displaystyle \tau }
G
2
k
{\displaystyle G_{2k}}
G
2
k
(
τ
)
=
∑
(
m
,
n
)
≠
(
0
,
0
)
1
(
m
+
n
τ
)
2
k
.
{\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
Γ
:=
S
L
(
2
,
Z
)
{\displaystyle \Gamma :=\mathrm {SL} (2,\mathbb {Z} )}
2
k
{\displaystyle 2k}
a
,
b
,
c
,
d
∈
Z
{\displaystyle a,b,c,d\in \mathbb {Z} }
a
d
−
b
c
=
1
{\displaystyle ad-bc=1}
G
2
k
(
a
τ
+
b
c
τ
+
d
)
=
(
c
τ
+
d
)
2
k
G
2
k
(
τ
)
{\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
Γ
{\displaystyle \Gamma }
G
4
{\displaystyle G_{4}}
G
6
{\displaystyle G_{6}}
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
G
4
,
G
6
{\displaystyle G_{4},G_{6}}
置
d
k
:=
(
2
k
+
3
)
k
!
G
2
k
+
4
{\displaystyle d_{k}:=(2k+3)k!G_{2k+4}}
∑
k
=
0
n
(
n
k
)
d
k
d
n
−
k
=
2
n
+
9
3
n
+
6
d
n
+
2
{\displaystyle \sum _{k=0}^{n}{n \choose k}d_{k}d_{n-k}={\frac {2n+9}{3n+6}}d_{n+2}}
在此
(
n
k
)
{\displaystyle {n \choose k}}
二項式系數 而
d
0
=
3
G
4
{\displaystyle d_{0}=3G_{4}}
d
1
=
5
G
6
{\displaystyle d_{1}=5G_{6}}
函數
d
k
{\displaystyle d_{k}}
魏爾斯特拉斯
℘
{\displaystyle \wp }
℘
(
z
)
=
1
z
2
+
z
2
∑
k
=
0
∞
d
k
z
2
k
k
!
=
1
z
2
+
∑
k
=
1
∞
(
2
k
+
1
)
G
2
k
+
2
z
2
k
{\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
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
G
2
k
(
τ
)
=
2
ζ
(
2
k
)
(
1
+
c
2
k
∑
n
=
1
∞
σ
2
k
−
1
(
n
)
q
n
)
{\displaystyle G_{2k}(\tau )=2\zeta (2k)\left(1+c_{2k}\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}\right)}
其中的傅立葉系數
c
2
k
{\displaystyle c_{2k}}
c
2
k
=
(
2
π
i
)
2
k
(
2
k
−
1
)
!
ζ
(
2
k
)
=
−
4
k
B
2
k
{\displaystyle c_{2k}={\frac {(2\pi i)^{2k}}{(2k-1)!\zeta (2k)}}={\frac {-4k}{B_{2k}}}}
此處的
B
n
{\displaystyle B_{n}}
伯努利數 ，
ζ
(
z
)
{\displaystyle \zeta (z)}
黎曼ζ函數 ，而
σ
p
(
n
)
{\displaystyle \sigma _{p}(n)}
n
{\displaystyle n}
因數 的
p
{\displaystyle p}
G
4
(
τ
)
=
π
4
45
[
1
+
240
∑
n
=
1
∞
σ
3
(
n
)
q
n
]
{\displaystyle G_{4}(\tau )={\frac {\pi ^{4}}{45}}\left[1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right]}
G
6
(
τ
)
=
2
π
6
945
[
1
−
504
∑
n
=
1
∞
σ
5
(
n
)
q
n
]
{\displaystyle G_{6}(\tau )={\frac {2\pi ^{6}}{945}}\left[1-504\sum _{n=1}^{\infty }\sigma _{5}(n)q^{n}\right]}
當
|
q
|
<
1
{\displaystyle |q|<1}
q
{\displaystyle q}
蘭伯特級數
∑
n
=
1
∞
q
n
σ
a
(
n
)
=
∑
n
=
1
∞
n
a
q
n
1
−
q
n
{\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：
E
2
k
:=
G
2
k
2
ζ
(
2
k
)
=
1
−
4
k
B
2
k
∑
n
=
1
∞
σ
2
k
−
1
(
n
)
q
n
{\displaystyle 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 給出了許多有趣的艾森斯坦級數關係式：定義
L
(
q
)
=
1
−
24
∑
n
=
1
∞
n
q
n
1
−
q
n
=
E
2
(
τ
)
{\displaystyle L(q)=1-24\sum _{n=1}^{\infty }{\frac {nq^{n}}{1-q^{n}}}=E_{2}(\tau )}
M
(
q
)
=
1
+
240
∑
n
=
1
∞
n
3
q
n
1
−
q
n
=
E
4
(
τ
)
{\displaystyle M(q)=1+240\sum _{n=1}^{\infty }{\frac {n^{3}q^{n}}{1-q^{n}}}=E_{4}(\tau )}
N
(
q
)
=
1
−
504
∑
n
=
1
∞
n
5
q
n
1
−
q
n
=
E
6
(
τ
)
{\displaystyle N(q)=1-504\sum _{n=1}^{\infty }{\frac {n^{5}q^{n}}{1-q^{n}}}=E_{6}(\tau )}
則有
q
d
L
d
q
=
L
2
−
M
12
{\displaystyle q{\frac {dL}{dq}}={\frac {L^{2}-M}{12}}}
q
d
M
d
q
=
L
M
−
N
3
{\displaystyle q{\frac {dM}{dq}}={\frac {LM-N}{3}}}
q
d
N
d
q
=
L
N
−
M
2
2
{\displaystyle q{\frac {dN}{dq}}={\frac {LN-M^{2}}{2}}}
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. 
![{\displaystyle G_{4}(\tau )={\frac {\pi ^{4}}{45}}\left[1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5902f599994a5c4abf9d138354c6c1954834021)
![{\displaystyle G_{6}(\tau )={\frac {2\pi ^{6}}{945}}\left[1-504\sum _{n=1}^{\infty }\sigma _{5}(n)q^{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2e3e0303beeb8b969d35a0ad2d66eb05c38acd)
