# Eisenstein series

## Definitions

Symbol: EisensteinG $G_{k}\!\left(\tau\right)$ Eisenstein series
Symbol: EisensteinE $E_{k}\!\left(\tau\right)$ Normalized Eisenstein series

## Illustrations

Related topics: Illustrations of Eisenstein series

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

## Normalization

$E_{2 k}\!\left(\tau\right) = \frac{G_{2 k}\!\left(\tau\right)}{2 \zeta\!\left(2 k\right)}$

## Lattice series

$G_{2 k}\!\left(\tau\right) = \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}$
$G_{2 k}\!\left(\tau\right) = \zeta\!\left(2 k\right) \sum_{\textstyle{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\} \atop \gcd\left(m, n\right) = 1}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}$
$G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right) + 2 \sum_{m=1}^{\infty} \sum_{n \in \mathbb{Z}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}$

## Modular transformations

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

### Quasi-modular transformations for weight 2

$G_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} G_{2}\!\left(\tau\right) - 2 \pi i c \left(c \tau + d\right)$
$E_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} E_{2}\!\left(\tau\right) - \frac{6 i}{\pi} c \left(c \tau + d\right)$

### Weight 2 quasi-holomorphic modular form

$H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H(\tau)\; \text{ where } H(\tau) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}(\tau)}$

## Fourier series (q-series)

### First cases

$E_{2}\!\left(\tau\right) = 1 - 24 \sum_{n=1}^{\infty} \sigma_{1}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}$
$E_{4}\!\left(\tau\right) = 1 + 240 \sum_{n=1}^{\infty} \sigma_{3}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}$
$E_{6}\!\left(\tau\right) = 1 - 504 \sum_{n=1}^{\infty} \sigma_{5}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}$

### General case

$E_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \sigma_{2 k - 1}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}$
$E_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \frac{{n}^{2 k - 1} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{2 \pi i \tau}$
$E_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} {n}^{2 k - 1} {q}^{m n}\; \text{ where } q = {e}^{2 \pi i \tau}$

## Trigonometric series

$E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}$
$E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}$
$E_{4}\!\left(\tau\right) = 1 + 30 \sum_{m=1}^{\infty} \frac{\cos^{2}\!\left(\pi m \tau\right) + 1}{\sin^{4}\!\left(\pi m \tau\right)}$
$E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}$

## Theta function representations

$E_{4}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)$
$E_{6}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{3}^{12}\!\left(0, \tau\right) + \theta_{4}^{12}\!\left(0, \tau\right) - 3 \theta_{2}^{8}\!\left(0, \tau\right) \left(\theta_{3}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)\right)\right)$
$E_{8}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{2}^{16}\!\left(0, \tau\right) + \theta_{3}^{16}\!\left(0, \tau\right) + \theta_{4}^{16}\!\left(0, \tau\right)\right)$
$E_{6}^{2}\!\left(\tau\right) = \frac{1}{8} \left({\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3} - 54 {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}\right)$
$E_{4}^{3}\!\left(\tau\right) - E_{6}^{2}\!\left(\tau\right) = \frac{27}{4} {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}$

## Dedekind eta function representations

$G_{2}\!\left(\tau\right) = -4 \pi i \frac{\eta'(\tau)}{\eta(\tau)}$
$E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}$
$E_{4}\!\left(\tau\right) = \frac{\eta^{16}\!\left(\tau\right)}{\eta^{8}\!\left(2 \tau\right)} + 256 \frac{\eta^{16}\!\left(2 \tau\right)}{\eta^{8}\!\left(\tau\right)}$
$E_{6}\!\left(\tau\right) = \frac{\eta^{24}\!\left(\tau\right)}{\eta^{12}\!\left(2 \tau\right)} - 480 \eta^{12}\!\left(2 \tau\right) - 16896 \frac{\eta^{12}\!\left(2 \tau\right) \eta^{8}\!\left(4 \tau\right)}{\eta^{8}\!\left(\tau\right)} + 8192 \frac{\eta^{24}\!\left(4 \tau\right)}{\eta^{12}\!\left(2 \tau\right)}$

## Elliptic function representations

$G_{2}\!\left(\tau\right) = 2 \zeta\!\left(\frac{1}{2}, \tau\right)$
$E_{2}\!\left(\tau\right) = \frac{6}{{\pi}^{2}} \zeta\!\left(\frac{1}{2}, \tau\right)$

## Products and recurrence relations

$E_{8}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right)$
$E_{10}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{6}\!\left(\tau\right)$
$E_{14}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right) E_{6}\!\left(\tau\right)$
$E_{14}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{10}\!\left(\tau\right)$
$E_{14}\!\left(\tau\right) = E_{6}\!\left(\tau\right) E_{8}\!\left(\tau\right)$
$E_{12}\!\left(\tau\right) = \frac{1}{691} \left(441 E_{4}^{3}\!\left(\tau\right) + 250 E_{6}^{2}\!\left(\tau\right)\right)$
$G_{2 k}\!\left(\tau\right) = \frac{3}{\left(2 k + 1\right) \left(k - 3\right) \left(2 k - 1\right)} \sum_{r=2}^{k - 2} \left(2 r - 1\right) \left(2 k - 2 r - 1\right) G_{2 r}\!\left(\tau\right) G_{2 k - 2 r}\!\left(\tau\right)$

## Generating functions

$\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}$
$\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}$

## Derivatives

$E'_{2}(\tau) = 2 \pi i \left(\frac{E_{2}^{2}\!\left(\tau\right) - E_{4}\!\left(\tau\right)}{12}\right)$
$E'_{4}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{4}\!\left(\tau\right) - E_{6}\!\left(\tau\right)}{3}\right)$
$E'_{6}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{6}\!\left(\tau\right) - E_{4}^{2}\!\left(\tau\right)}{2}\right)$

## Specific values

### Fourth root of unity

$G_{2}\!\left(i\right) = \pi$
$E_{2}\!\left(i\right) = \frac{3}{\pi}$
$G_{4}\!\left(i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{960 {\pi}^{2}}$
$E_{4}\!\left(i\right) = \frac{3 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{64 {\pi}^{6}}$
$G_{6}\!\left(i\right) = E_{6}\!\left(i\right) = 0$

### Third root of unity

$G_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \pi}{\sqrt{3}}$
$E_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \sqrt{3}}{\pi}$
$G_{4}\!\left({e}^{2 \pi i / 3}\right) = E_{4}\!\left({e}^{2 \pi i / 3}\right) = 0$
$G_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{8960 {\pi}^{6}}$
$E_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{27 {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{512 {\pi}^{12}}$

### Infinity

$G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right)$
$E_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} E_{2 k}\!\left(\tau\right) = 1$

## Zeros

### Distribution

$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \subset \left\{ {e}^{i \theta} : \theta \in \left[\frac{\pi}{2}, \frac{2 \pi}{3}\right] \right\}$
$\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1$
$\sum_{\tau \in \mathcal{F}} w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = \frac{2 k}{12}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\\frac{1}{3}, & \tau = {e}^{2 \pi i / 3}\\1, & \text{otherwise}\\ \end{cases}$
$\sum_{\tau \in \mathcal{F}} j(\tau) w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\1, & \text{otherwise}\\ \end{cases}$

### Specific values

$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{4}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{6}\!\left(\tau\right) = \left\{i\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{8}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{10}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{12}\!\left(\tau\right) = \left\{\frac{i \,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, a\right)}{\,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, 1 - a\right)}\right\}\; \text{ where } a = \frac{1}{2} + \frac{21 \sqrt{10} i}{100}$