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Eisenstein series

Table of contents: Definitions - Illustrations - Normalization - Lattice series - Modular transformations - Fourier series (q-series) - Trigonometric series - Theta function representations - Dedekind eta function representations - Elliptic function representations - Products and recurrence relations - Generating functions - Derivatives - Specific values - Zeros

Definitions

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Symbol: EisensteinG Gk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
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Symbol: EisensteinE Ek ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series

Illustrations

Related topics: Illustrations of Eisenstein series

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Image: X-ray of E6 ⁣(τ)E_{6}\!\left(\tau\right) on τ[1,1]+[0,2]i\tau \in \left[-1, 1\right] + \left[0, 2\right] i with F\mathcal{F} highlighted

Normalization

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E2k ⁣(τ)=G2k ⁣(τ)2ζ ⁣(2k)E_{2 k}\!\left(\tau\right) = \frac{G_{2 k}\!\left(\tau\right)}{2 \zeta\!\left(2 k\right)}

Lattice series

2246a7
G2k ⁣(τ)=(m,n)Z2{(0,0)}1(mτ+n)2kG_{2 k}\!\left(\tau\right) = \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}
c1ffd4
G2k ⁣(τ)=ζ ⁣(2k)(m,n)Z2{(0,0)}gcd(m,n)=11(mτ+n)2kG_{2 k}\!\left(\tau\right) = \zeta\!\left(2 k\right) \sum_{\textstyle{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\} \atop \gcd\left(m, n\right) = 1}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}
b07750
G2k ⁣(τ)=2ζ ⁣(2k)+2m=1nZ1(mτ+n)2kG_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right) + 2 \sum_{m=1}^{\infty} \sum_{n \in \mathbb{Z}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}

Modular transformations

0b5b04
G2k ⁣(aτ+bcτ+d)=(cτ+d)2kG2k ⁣(τ)G_{2 k}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2 k} G_{2 k}\!\left(\tau\right)
8ffe07
E2k ⁣(aτ+bcτ+d)=(cτ+d)2kE2k ⁣(τ)E_{2 k}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2 k} E_{2 k}\!\left(\tau\right)
23a5e0
G2k ⁣(τ+n)=G2k ⁣(τ)G_{2 k}\!\left(\tau + n\right) = G_{2 k}\!\left(\tau\right)
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E2k ⁣(τ+n)=E2k ⁣(τ)E_{2 k}\!\left(\tau + n\right) = E_{2 k}\!\left(\tau\right)

Quasi-modular transformations for weight 2

5161ab
G2 ⁣(aτ+bcτ+d)=(cτ+d)2G2 ⁣(τ)2πic(cτ+d)G_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} G_{2}\!\left(\tau\right) - 2 \pi i c \left(c \tau + d\right)
7f4c85
E2 ⁣(aτ+bcτ+d)=(cτ+d)2E2 ⁣(τ)6iπc(cτ+d)E_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} E_{2}\!\left(\tau\right) - \frac{6 i}{\pi} c \left(c \tau + d\right)

Weight 2 quasi-holomorphic modular form

b1a5e4
H ⁣(aτ+bcτ+d)=(cτ+d)2H(τ)   where H(τ)=G2 ⁣(τ)πIm(τ)H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H(\tau)\; \text{ where } H(\tau) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}(\tau)}

Fourier series (q-series)

First cases

10cdf4
E2 ⁣(τ)=124n=1σ1 ⁣(n)qn   where q=e2πiτE_{2}\!\left(\tau\right) = 1 - 24 \sum_{n=1}^{\infty} \sigma_{1}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}
f8dfaf
E4 ⁣(τ)=1+240n=1σ3 ⁣(n)qn   where q=e2πiτE_{4}\!\left(\tau\right) = 1 + 240 \sum_{n=1}^{\infty} \sigma_{3}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}
e20db0
E6 ⁣(τ)=1504n=1σ5 ⁣(n)qn   where q=e2πiτE_{6}\!\left(\tau\right) = 1 - 504 \sum_{n=1}^{\infty} \sigma_{5}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}

General case

7c00e6
E2k ⁣(τ)=14nB2nn=1σ2k1 ⁣(n)qn   where q=e2πiτE_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \sigma_{2 k - 1}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}
848d97
E2k ⁣(τ)=14nB2nn=1n2k1qn1qn   where q=e2πiτE_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \frac{{n}^{2 k - 1} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{2 \pi i \tau}
15b347
E2k ⁣(τ)=14nB2nn=1m=1n2k1qmn   where q=e2πiτE_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} {n}^{2 k - 1} {q}^{m n}\; \text{ where } q = {e}^{2 \pi i \tau}

Trigonometric series

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E2 ⁣(τ)=1+6m=11sin2 ⁣(πmτ)E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}
7b62e4
E2 ⁣(τ)=112m=11cos ⁣(2πmτ)1E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}
a92c1a
E4 ⁣(τ)=1+30m=1cos2 ⁣(πmτ)+1sin4 ⁣(πmτ)E_{4}\!\left(\tau\right) = 1 + 30 \sum_{m=1}^{\infty} \frac{\cos^{2}\!\left(\pi m \tau\right) + 1}{\sin^{4}\!\left(\pi m \tau\right)}
171724
E6 ⁣(τ)=1+63m=12cos4 ⁣(πmτ)+11cos2 ⁣(πmτ)+2sin6 ⁣(πmτ)E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}

Theta function representations

cc579c
E4 ⁣(τ)=12(θ28 ⁣(0,τ)+θ38 ⁣(0,τ)+θ48 ⁣(0,τ))E_{4}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)
10f3b2
E6 ⁣(τ)=12(θ312 ⁣(0,τ)+θ412 ⁣(0,τ)3θ28 ⁣(0,τ)(θ34 ⁣(0,τ)+θ44 ⁣(0,τ)))E_{6}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{3}^{12}\!\left(0, \tau\right) + \theta_{4}^{12}\!\left(0, \tau\right) - 3 \theta_{2}^{8}\!\left(0, \tau\right) \left(\theta_{3}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)\right)\right)
6d2880
E8 ⁣(τ)=12(θ216 ⁣(0,τ)+θ316 ⁣(0,τ)+θ416 ⁣(0,τ))E_{8}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{2}^{16}\!\left(0, \tau\right) + \theta_{3}^{16}\!\left(0, \tau\right) + \theta_{4}^{16}\!\left(0, \tau\right)\right)
a0dff6
E62 ⁣(τ)=18((θ28 ⁣(0,τ)+θ38 ⁣(0,τ)+θ48 ⁣(0,τ))354(θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ))8)E_{6}^{2}\!\left(\tau\right) = \frac{1}{8} \left({\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3} - 54 {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}\right)
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E43 ⁣(τ)E62 ⁣(τ)=274(θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ))8E_{4}^{3}\!\left(\tau\right) - E_{6}^{2}\!\left(\tau\right) = \frac{27}{4} {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}

Dedekind eta function representations

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G2 ⁣(τ)=4πiη(τ)η(τ)G_{2}\!\left(\tau\right) = -4 \pi i \frac{\eta'(\tau)}{\eta(\tau)}
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E2 ⁣(τ)=12iπη(τ)η(τ)E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}
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E4 ⁣(τ)=η16 ⁣(τ)η8 ⁣(2τ)+256η16 ⁣(2τ)η8 ⁣(τ)E_{4}\!\left(\tau\right) = \frac{\eta^{16}\!\left(\tau\right)}{\eta^{8}\!\left(2 \tau\right)} + 256 \frac{\eta^{16}\!\left(2 \tau\right)}{\eta^{8}\!\left(\tau\right)}
0a5ef4
E6 ⁣(τ)=η24 ⁣(τ)η12 ⁣(2τ)480η12 ⁣(2τ)16896η12 ⁣(2τ)η8 ⁣(4τ)η8 ⁣(τ)+8192η24 ⁣(4τ)η12 ⁣(2τ)E_{6}\!\left(\tau\right) = \frac{\eta^{24}\!\left(\tau\right)}{\eta^{12}\!\left(2 \tau\right)} - 480 \eta^{12}\!\left(2 \tau\right) - 16896 \frac{\eta^{12}\!\left(2 \tau\right) \eta^{8}\!\left(4 \tau\right)}{\eta^{8}\!\left(\tau\right)} + 8192 \frac{\eta^{24}\!\left(4 \tau\right)}{\eta^{12}\!\left(2 \tau\right)}

Elliptic function representations

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G2 ⁣(τ)=2ζ ⁣(12,τ)G_{2}\!\left(\tau\right) = 2 \zeta\!\left(\frac{1}{2}, \tau\right)
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E2 ⁣(τ)=6π2ζ ⁣(12,τ)E_{2}\!\left(\tau\right) = \frac{6}{{\pi}^{2}} \zeta\!\left(\frac{1}{2}, \tau\right)

Products and recurrence relations

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E8 ⁣(τ)=E42 ⁣(τ)E_{8}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right)
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E10 ⁣(τ)=E4 ⁣(τ)E6 ⁣(τ)E_{10}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{6}\!\left(\tau\right)
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E14 ⁣(τ)=E42 ⁣(τ)E6 ⁣(τ)E_{14}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right) E_{6}\!\left(\tau\right)
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E14 ⁣(τ)=E4 ⁣(τ)E10 ⁣(τ)E_{14}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{10}\!\left(\tau\right)
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E14 ⁣(τ)=E6 ⁣(τ)E8 ⁣(τ)E_{14}\!\left(\tau\right) = E_{6}\!\left(\tau\right) E_{8}\!\left(\tau\right)
36fff2
E12 ⁣(τ)=1691(441E43 ⁣(τ)+250E62 ⁣(τ))E_{12}\!\left(\tau\right) = \frac{1}{691} \left(441 E_{4}^{3}\!\left(\tau\right) + 250 E_{6}^{2}\!\left(\tau\right)\right)
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G2k ⁣(τ)=3(2k+1)(k3)(2k1)r=2k2(2r1)(2k2r1)G2r ⁣(τ)G2k2r ⁣(τ)G_{2 k}\!\left(\tau\right) = \frac{3}{\left(2 k + 1\right) \left(k - 3\right) \left(2 k - 1\right)} \sum_{r=2}^{k - 2} \left(2 r - 1\right) \left(2 k - 2 r - 1\right) G_{2 r}\!\left(\tau\right) G_{2 k - 2 r}\!\left(\tau\right)

Generating functions

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 ⁣(z,τ)=1z2+k=1(2k+1)G2k+2 ⁣(τ)z2k\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}
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ζ ⁣(z,τ)=1zk=1G2k+2 ⁣(τ)z2k+1\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}

Derivatives

7cda09
E2(τ)=2πi(E22 ⁣(τ)E4 ⁣(τ)12)E'_{2}(\tau) = 2 \pi i \left(\frac{E_{2}^{2}\!\left(\tau\right) - E_{4}\!\left(\tau\right)}{12}\right)
af2ea9
E4(τ)=2πi(E2 ⁣(τ)E4 ⁣(τ)E6 ⁣(τ)3)E'_{4}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{4}\!\left(\tau\right) - E_{6}\!\left(\tau\right)}{3}\right)
3bfced
E6(τ)=2πi(E2 ⁣(τ)E6 ⁣(τ)E42 ⁣(τ)2)E'_{6}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{6}\!\left(\tau\right) - E_{4}^{2}\!\left(\tau\right)}{2}\right)

Specific values

Fourth root of unity

570399
G2 ⁣(i)=πG_{2}\!\left(i\right) = \pi
a691b3
E2 ⁣(i)=3πE_{2}\!\left(i\right) = \frac{3}{\pi}
e03b7c
G4 ⁣(i)=(Γ ⁣(14))8960π2G_{4}\!\left(i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{960 {\pi}^{2}}
53fcdd
E4 ⁣(i)=3(Γ ⁣(14))864π6E_{4}\!\left(i\right) = \frac{3 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{64 {\pi}^{6}}
a4109c
G6 ⁣(i)=E6 ⁣(i)=0G_{6}\!\left(i\right) = E_{6}\!\left(i\right) = 0

Third root of unity

9ea739
G2 ⁣(e2πi/3)=2π3G_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \pi}{\sqrt{3}}
30a054
E2 ⁣(e2πi/3)=23πE_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \sqrt{3}}{\pi}
3102a7
G4 ⁣(e2πi/3)=E4 ⁣(e2πi/3)=0G_{4}\!\left({e}^{2 \pi i / 3}\right) = E_{4}\!\left({e}^{2 \pi i / 3}\right) = 0
0fda1b
G6 ⁣(e2πi/3)=(Γ ⁣(13))188960π6G_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{8960 {\pi}^{6}}
6c71c0
E6 ⁣(e2πi/3)=27(Γ ⁣(13))18512π12E_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{27 {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{512 {\pi}^{12}}

Infinity

c6be24
G2k ⁣(i)=limτiG2k ⁣(τ)=2ζ ⁣(2k)G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right)
ad9ba2
E2k ⁣(i)=limτiE2k ⁣(τ)=1E_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} E_{2 k}\!\left(\tau\right) = 1

Zeros

Distribution

e46697
zerosτHE2k ⁣(τ)={γτ:τzeroszFE2k ⁣(z)andγPSL2(Z)}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}
2f6805
zerosτFE2k ⁣(τ){eiθ:θ[π2,2π3]}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \subset \left\{ {e}^{i \theta} : \theta \in \left[\frac{\pi}{2}, \frac{2 \pi}{3}\right] \right\}
a50278
#zerosτFE2k ⁣(τ)1\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1
13cac5
τFw(τ)ordz=τE2k ⁣(z)=2k12   where w(τ)={12,τ=i13,τ=e2πi/31,otherwise\sum_{\tau \in \mathcal{F}} w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = \frac{2 k}{12}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\\frac{1}{3}, & \tau = {e}^{2 \pi i / 3}\\1, & \text{otherwise}\\ \end{cases}
097efc
τFj(τ)w(τ)ordz=τE2k ⁣(z)=120k2ζ ⁣(12k)   where w(τ)={12,τ=i1,otherwise\sum_{\tau \in \mathcal{F}} j(\tau) w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\1, & \text{otherwise}\\ \end{cases}

Specific values

4a200a
zerosτFE4 ⁣(τ)={e2πi/3}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{4}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}
ec4f56
zerosτFE6 ⁣(τ)={i}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{6}\!\left(\tau\right) = \left\{i\right\}
83566f
zerosτFE8 ⁣(τ)={e2πi/3}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{8}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}
26faf3
zerosτFE10 ⁣(τ)={i,e2πi/3}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{10}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}
6ae250
zerosτFE12 ⁣(τ)={i2F1 ⁣(16,56,1,a)2F1 ⁣(16,56,1,1a)}   where a=12+2110i100\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{12}\!\left(\tau\right) = \left\{\frac{i \,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, a\right)}{\,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, 1 - a\right)}\right\}\; \text{ where } a = \frac{1}{2} + \frac{21 \sqrt{10} i}{100}