Eisenstein series

Symbol: EisensteinG —

G_{k}\!\left(\tau\right)

— Eisenstein series

The Eisenstein series

G_{k}\!\left(\tau\right)

is defined for even integers

k \ge 2

and

\tau

in the upper half-plane.

The functions

G_{k}\!\left(\tau\right)

and

E_{k}\!\left(\tau\right)

are the same up to a normalizing factor.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series

Source code for this entry:

Entry(ID("9bb960"),
    SymbolDefinition(EisensteinG, EisensteinG(k, tau), "Eisenstein series"),
    Description("The Eisenstein series", EisensteinG(k, tau), "is defined for even integers", GreaterEqual(k, 2), "and", tau, "in the upper half-plane."),
    Description("The functions", EisensteinG(k, tau), "and", EisensteinE(k, tau), "are the same up to a normalizing factor."))

df3334

Symbol: EisensteinE —

E_{k}\!\left(\tau\right)

— Normalized Eisenstein series

The normalized Eisenstein series

E_{k}\!\left(\tau\right)

is defined for even integers

k \ge 2

and

\tau

in the upper half-plane.

The functions

G_{k}\!\left(\tau\right)

and

E_{k}\!\left(\tau\right)

are the same up to a normalizing factor.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series

Source code for this entry:

Entry(ID("df3334"),
    SymbolDefinition(EisensteinE, EisensteinE(k, tau), "Normalized Eisenstein series"),
    Description("The normalized Eisenstein series", EisensteinE(k, tau), "is defined for even integers", GreaterEqual(k, 2), "and", tau, "in the upper half-plane."),
    Description("The functions", EisensteinG(k, tau), "and", EisensteinE(k, tau), "are the same up to a normalizing factor."))

54951a

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

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("54951a"),
    Image(Description("X-ray of", EisensteinE(6, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e6")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

0a2120

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

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0a2120"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Div(EisensteinG(Mul(2, k), tau), Mul(2, RiemannZeta(Mul(2, k)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

2246a7

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}}

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}}

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2246a7"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Sum(Div(1, Pow(Add(Mul(m, tau), n), Mul(2, k))), ForElement(Tuple(m, n), SetMinus(Pow(ZZ, 2), Set(Tuple(0, 0))))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH))))

c1ffd4

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}}

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}}

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("c1ffd4"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Mul(RiemannZeta(Mul(2, k)), Sum(Div(1, Pow(Add(Mul(m, tau), n), Mul(2, k))), ForElement(Tuple(m, n), SetMinus(Pow(ZZ, 2), Set(Tuple(0, 0)))), Equal(GCD(m, n), 1))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH))))

b07750

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}}

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("b07750"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Add(Mul(2, RiemannZeta(Mul(2, k))), Mul(2, Sum(Sum(Div(1, Pow(Add(Mul(m, tau), n), Mul(2, k))), ForElement(n, ZZ)), For(m, 1, Infinity)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

0b5b04

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)

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

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)

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("0b5b04"),
    Formula(Equal(EisensteinG(Mul(2, k), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), Mul(2, k)), EisensteinG(Mul(2, k), tau)))),
    Variables(k, tau, a, b, c, d),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

8ffe07

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)

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

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)

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("8ffe07"),
    Formula(Equal(EisensteinE(Mul(2, k), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), Mul(2, k)), EisensteinE(Mul(2, k), tau)))),
    Variables(k, tau, a, b, c, d),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

23a5e0

G_{2 k}\!\left(\tau + n\right) = G_{2 k}\!\left(\tau\right)

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

G_{2 k}\!\left(\tau + n\right) = G_{2 k}\!\left(\tau\right)

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("23a5e0"),
    Formula(Equal(EisensteinG(Mul(2, k), Add(tau, n)), EisensteinG(Mul(2, k), tau))),
    Variables(k, n, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH), Element(n, ZZ))))

d56eb6

E_{2 k}\!\left(\tau + n\right) = E_{2 k}\!\left(\tau\right)

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

E_{2 k}\!\left(\tau + n\right) = E_{2 k}\!\left(\tau\right)

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d56eb6"),
    Formula(Equal(EisensteinE(Mul(2, k), Add(tau, n)), EisensteinE(Mul(2, k), tau))),
    Variables(k, n, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH), Element(n, ZZ))))

5161ab

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)

Assumptions:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

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)

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("5161ab"),
    Formula(Equal(EisensteinG(2, Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Sub(Mul(Pow(Add(Mul(c, tau), d), 2), EisensteinG(2, tau)), Mul(Mul(Mul(Mul(2, Pi), ConstI), c), Add(Mul(c, tau), d))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

7f4c85

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)

Assumptions:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

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)

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("7f4c85"),
    Formula(Equal(EisensteinE(2, Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Sub(Mul(Pow(Add(Mul(c, tau), d), 2), EisensteinE(2, tau)), Mul(Mul(Div(Mul(6, ConstI), Pi), c), Add(Mul(c, tau), d))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

b1a5e4

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)}

Assumptions:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

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)}

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`Im`	$\operatorname{Im}(z)$	Imaginary part
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("b1a5e4"),
    Formula(Where(Equal(H(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), 2), H(tau))), Equal(H(tau), Sub(EisensteinG(2, tau), Div(Pi, Im(tau)))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

10cdf4

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}

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("10cdf4"),
    Formula(Equal(EisensteinE(2, tau), Where(Sub(1, Mul(24, Sum(Mul(DivisorSigma(1, n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

f8dfaf

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}

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f8dfaf"),
    Formula(Equal(EisensteinE(4, tau), Where(Add(1, Mul(240, Sum(Mul(DivisorSigma(3, n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

e20db0

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}

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("e20db0"),
    Formula(Equal(EisensteinE(6, tau), Where(Sub(1, Mul(504, Sum(Mul(DivisorSigma(5, n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

7c00e6

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

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`BernoulliB`	$B_{n}$	Bernoulli number
`Sum`	$\sum_{n} f(n)$	Sum
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7c00e6"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Mul(DivisorSigma(Sub(Mul(2, k), 1), n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

848d97

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

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`BernoulliB`	$B_{n}$	Bernoulli number
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("848d97"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Div(Mul(Pow(n, Sub(Mul(2, k), 1)), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

15b347

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

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

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

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`BernoulliB`	$B_{n}$	Bernoulli number
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("15b347"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Sum(Mul(Pow(n, Sub(Mul(2, k), 1)), Pow(q, Mul(m, n))), For(m, 1, Infinity)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

18a4d1

E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("18a4d1"),
    Formula(Equal(EisensteinE(2, tau), Add(1, Mul(6, Sum(Div(1, Pow(Sin(Mul(Mul(Pi, m), tau)), 2)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

7b62e4

E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7b62e4"),
    Formula(Equal(EisensteinE(2, tau), Sub(1, Mul(12, Sum(Div(1, Sub(Cos(Mul(Mul(Mul(2, Pi), m), tau)), 1)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

a92c1a

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)}

Assumptions:

\tau \in \mathbb{H}

TeX:

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)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a92c1a"),
    Formula(Equal(EisensteinE(4, tau), Add(1, Mul(30, Sum(Div(Add(Pow(Cos(Mul(Mul(Pi, m), tau)), 2), 1), Pow(Sin(Mul(Mul(Pi, m), tau)), 4)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

171724

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)}

Assumptions:

\tau \in \mathbb{H}

TeX:

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)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("171724"),
    Formula(Equal(EisensteinE(6, tau), Add(1, Mul(63, Sum(Div(Add(Add(Mul(2, Pow(Cos(Mul(Mul(Pi, m), tau)), 4)), Mul(11, Pow(Cos(Mul(Mul(Pi, m), tau)), 2))), 2), Pow(Sin(Mul(Mul(Pi, m), tau)), 6)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

cc579c

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)

Assumptions:

\tau \in \mathbb{H}

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("cc579c"),
    Formula(Equal(EisensteinE(4, tau), Mul(Div(1, 2), Add(Add(Pow(JacobiTheta(2, 0, tau), 8), Pow(JacobiTheta(3, 0, tau), 8)), Pow(JacobiTheta(4, 0, tau), 8))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

10f3b2

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)

Assumptions:

\tau \in \mathbb{H}

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("10f3b2"),
    Formula(Equal(EisensteinE(6, tau), Mul(Div(1, 2), Sub(Add(Pow(JacobiTheta(3, 0, tau), 12), Pow(JacobiTheta(4, 0, tau), 12)), Mul(Mul(3, Pow(JacobiTheta(2, 0, tau), 8)), Add(Pow(JacobiTheta(3, 0, tau), 4), Pow(JacobiTheta(4, 0, tau), 4))))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

6d2880

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)

Assumptions:

\tau \in \mathbb{H}

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("6d2880"),
    Formula(Equal(EisensteinE(8, tau), Mul(Div(1, 2), Add(Add(Pow(JacobiTheta(2, 0, tau), 16), Pow(JacobiTheta(3, 0, tau), 16)), Pow(JacobiTheta(4, 0, tau), 16))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

a0dff6

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)

Assumptions:

\tau \in \mathbb{H}

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a0dff6"),
    Formula(Equal(Pow(EisensteinE(6, tau), 2), Mul(Div(1, 8), Sub(Pow(Add(Add(Pow(JacobiTheta(2, 0, tau), 8), Pow(JacobiTheta(3, 0, tau), 8)), Pow(JacobiTheta(4, 0, tau), 8)), 3), Mul(54, Pow(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(4, 0, tau)), 8)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

bd7d8e

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}

Assumptions:

\tau \in \mathbb{H}

TeX:

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}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("bd7d8e"),
    Formula(Equal(Sub(Pow(EisensteinE(4, tau), 3), Pow(EisensteinE(6, tau), 2)), Mul(Div(27, 4), Pow(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(4, 0, tau)), 8)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

dbf388

G_{2}\!\left(\tau\right) = -4 \pi i \frac{\eta'(\tau)}{\eta(\tau)}

Assumptions:

\tau \in \mathbb{H}

TeX:

G_{2}\!\left(\tau\right) = -4 \pi i \frac{\eta'(\tau)}{\eta(\tau)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("dbf388"),
    Formula(Equal(EisensteinG(2, tau), Neg(Mul(Mul(Mul(4, Pi), ConstI), Div(ComplexDerivative(DedekindEta(tau), For(tau, tau)), DedekindEta(tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

03ad5a

E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("03ad5a"),
    Formula(Equal(EisensteinE(2, tau), Neg(Mul(Div(Mul(12, ConstI), Pi), Div(ComplexDerivative(DedekindEta(tau), For(tau, tau)), DedekindEta(tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

4da2cd

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)}

Assumptions:

\tau \in \mathbb{H}

References:

K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67.

TeX:

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)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("4da2cd"),
    Formula(Equal(EisensteinE(4, tau), Add(Div(Pow(DedekindEta(tau), 16), Pow(DedekindEta(Mul(2, tau)), 8)), Mul(256, Div(Pow(DedekindEta(Mul(2, tau)), 16), Pow(DedekindEta(tau), 8)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67."))

0a5ef4

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)}

Assumptions:

\tau \in \mathbb{H}

References:

K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67.

TeX:

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)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0a5ef4"),
    Formula(Equal(EisensteinE(6, tau), Add(Sub(Sub(Div(Pow(DedekindEta(tau), 24), Pow(DedekindEta(Mul(2, tau)), 12)), Mul(480, Pow(DedekindEta(Mul(2, tau)), 12))), Mul(16896, Div(Mul(Pow(DedekindEta(Mul(2, tau)), 12), Pow(DedekindEta(Mul(4, tau)), 8)), Pow(DedekindEta(tau), 8)))), Mul(8192, Div(Pow(DedekindEta(Mul(4, tau)), 24), Pow(DedekindEta(Mul(2, tau)), 12)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67."))

b52b6f

G_{2}\!\left(\tau\right) = 2 \zeta\!\left(\frac{1}{2}, \tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

G_{2}\!\left(\tau\right) = 2 \zeta\!\left(\frac{1}{2}, \tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("b52b6f"),
    Formula(Equal(EisensteinG(2, tau), Mul(2, WeierstrassZeta(Div(1, 2), tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

3bf702

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

Assumptions:

\tau \in \mathbb{H}

TeX:

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

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3bf702"),
    Formula(Equal(EisensteinE(2, tau), Mul(Div(6, Pow(Pi, 2)), WeierstrassZeta(Div(1, 2), tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

044128

E_{8}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{8}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("044128"),
    Formula(Equal(EisensteinE(8, tau), Pow(EisensteinE(4, tau), 2))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

adaf5a

E_{10}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{6}\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{10}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{6}\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("adaf5a"),
    Formula(Equal(EisensteinE(10, tau), Mul(EisensteinE(4, tau), EisensteinE(6, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

e60fd4

E_{14}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right) E_{6}\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{14}\!\left(\tau\right) = E_{4}^{2}\!\left(\tau\right) E_{6}\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("e60fd4"),
    Formula(Equal(EisensteinE(14, tau), Mul(Pow(EisensteinE(4, tau), 2), EisensteinE(6, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

9e1f83

E_{14}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{10}\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{14}\!\left(\tau\right) = E_{4}\!\left(\tau\right) E_{10}\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("9e1f83"),
    Formula(Equal(EisensteinE(14, tau), Mul(EisensteinE(4, tau), EisensteinE(10, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

feb95e

E_{14}\!\left(\tau\right) = E_{6}\!\left(\tau\right) E_{8}\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{14}\!\left(\tau\right) = E_{6}\!\left(\tau\right) E_{8}\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("feb95e"),
    Formula(Equal(EisensteinE(14, tau), Mul(EisensteinE(6, tau), EisensteinE(8, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

36fff2

E_{12}\!\left(\tau\right) = \frac{1}{691} \left(441 E_{4}^{3}\!\left(\tau\right) + 250 E_{6}^{2}\!\left(\tau\right)\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{12}\!\left(\tau\right) = \frac{1}{691} \left(441 E_{4}^{3}\!\left(\tau\right) + 250 E_{6}^{2}\!\left(\tau\right)\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("36fff2"),
    Formula(Equal(EisensteinE(12, tau), Mul(Div(1, 691), Add(Mul(441, Pow(EisensteinE(4, tau), 3)), Mul(250, Pow(EisensteinE(6, tau), 2)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

5540a1

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)

Assumptions:

k \in \mathbb{Z}_{\ge 4} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

References:

T. Apostol (1976), Modular Functions and Dirichlet Series in Number Theory, Springer. Theorem 1.13.

TeX:

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)

k \in \mathbb{Z}_{\ge 4} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("5540a1"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Mul(Div(3, Mul(Mul(Add(Mul(2, k), 1), Sub(k, 3)), Sub(Mul(2, k), 1))), Sum(Mul(Mul(Mul(Sub(Mul(2, r), 1), Sub(Sub(Mul(2, k), Mul(2, r)), 1)), EisensteinG(Mul(2, r), tau)), EisensteinG(Sub(Mul(2, k), Mul(2, r)), tau)), For(r, 2, Sub(k, 2)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(4)), Element(tau, HH))),
    References("T. Apostol (1976), Modular Functions and Dirichlet Series in Number Theory, Springer. Theorem 1.13."))

9bf0ad

\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}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}

TeX:

\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}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`WeierstrassP`	$\wp\!\left(z, \tau\right)$	Weierstrass elliptic function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Infimum`	$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$	Infimum of a set or function
`Lattice`	$\Lambda_{(a, b)}$	Complex lattice with periods a, b

Source code for this entry:

Entry(ID("9bf0ad"),
    Formula(Equal(WeierstrassP(z, tau), Add(Div(1, Pow(z, 2)), Sum(Mul(Mul(Add(Mul(2, k), 1), EisensteinG(Add(Mul(2, k), 2), tau)), Pow(z, Mul(2, k))), For(k, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(z), Infimum(Set(Abs(s), ForElement(s, SetMinus(Lattice(1, tau), Set(0)))))))))

3e84e3

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|z\right| < \operatorname{inf} \left\{ \left|s\right| : s \in \Lambda_{(1, \tau)} \setminus \left\{0\right\} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Infimum`	$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$	Infimum of a set or function
`Lattice`	$\Lambda_{(a, b)}$	Complex lattice with periods a, b

Source code for this entry:

Entry(ID("3e84e3"),
    Formula(Equal(WeierstrassZeta(z, tau), Sub(Div(1, z), Sum(Mul(EisensteinG(Add(Mul(2, k), 2), tau), Pow(z, Add(Mul(2, k), 1))), For(k, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(z), Infimum(Set(Abs(s), ForElement(s, SetMinus(Lattice(1, tau), Set(0)))))))))

7cda09

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

Assumptions:

\tau \in \mathbb{H}

References:

B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3

TeX:

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

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7cda09"),
    Formula(Equal(ComplexDerivative(EisensteinE(2, tau), For(tau, tau)), Mul(Mul(Mul(2, Pi), ConstI), Parentheses(Div(Sub(Pow(EisensteinE(2, tau), 2), EisensteinE(4, tau)), 12))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3"))

af2ea9

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)

Assumptions:

\tau \in \mathbb{H}

References:

B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("af2ea9"),
    Formula(Equal(ComplexDerivative(EisensteinE(4, tau), For(tau, tau)), Mul(Mul(Mul(2, Pi), ConstI), Parentheses(Div(Sub(Mul(EisensteinE(2, tau), EisensteinE(4, tau)), EisensteinE(6, tau)), 3))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3"))

3bfced

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)

Assumptions:

\tau \in \mathbb{H}

References:

B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3

TeX:

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)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3bfced"),
    Formula(Equal(ComplexDerivative(EisensteinE(6, tau), For(tau, tau)), Mul(Mul(Mul(2, Pi), ConstI), Parentheses(Div(Sub(Mul(EisensteinE(2, tau), EisensteinE(6, tau)), Pow(EisensteinE(4, tau), 2)), 2))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3"))

570399

G_{2}\!\left(i\right) = \pi

TeX:

G_{2}\!\left(i\right) = \pi

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("570399"),
    Formula(Equal(EisensteinG(2, ConstI), Pi)))

a691b3

E_{2}\!\left(i\right) = \frac{3}{\pi}

TeX:

E_{2}\!\left(i\right) = \frac{3}{\pi}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("a691b3"),
    Formula(Equal(EisensteinE(2, ConstI), Div(3, Pi))))

e03b7c

G_{4}\!\left(i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{960 {\pi}^{2}}

TeX:

G_{4}\!\left(i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{8}}{960 {\pi}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("e03b7c"),
    Formula(Equal(EisensteinG(4, ConstI), Div(Pow(Gamma(Div(1, 4)), 8), Mul(960, Pow(Pi, 2))))))

53fcdd

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

TeX:

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

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("53fcdd"),
    Formula(Equal(EisensteinE(4, ConstI), Div(Mul(3, Pow(Gamma(Div(1, 4)), 8)), Mul(64, Pow(Pi, 6))))))

a4109c

G_{6}\!\left(i\right) = E_{6}\!\left(i\right) = 0

TeX:

G_{6}\!\left(i\right) = E_{6}\!\left(i\right) = 0

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ConstI`	$i$	Imaginary unit
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series

Source code for this entry:

Entry(ID("a4109c"),
    Formula(Equal(EisensteinG(6, ConstI), EisensteinE(6, ConstI), 0)))

9ea739

G_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \pi}{\sqrt{3}}

TeX:

G_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \pi}{\sqrt{3}}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("9ea739"),
    Formula(Equal(EisensteinG(2, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), Div(Mul(2, Pi), Sqrt(3)))))

30a054

E_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \sqrt{3}}{\pi}

TeX:

E_{2}\!\left({e}^{2 \pi i / 3}\right) = \frac{2 \sqrt{3}}{\pi}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("30a054"),
    Formula(Equal(EisensteinE(2, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), Div(Mul(2, Sqrt(3)), Pi))))

3102a7

G_{4}\!\left({e}^{2 \pi i / 3}\right) = E_{4}\!\left({e}^{2 \pi i / 3}\right) = 0

TeX:

G_{4}\!\left({e}^{2 \pi i / 3}\right) = E_{4}\!\left({e}^{2 \pi i / 3}\right) = 0

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series

Source code for this entry:

Entry(ID("3102a7"),
    Formula(Equal(EisensteinG(4, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), EisensteinE(4, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), 0)))

0fda1b

G_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{8960 {\pi}^{6}}

TeX:

G_{6}\!\left({e}^{2 \pi i / 3}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{18}}{8960 {\pi}^{6}}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("0fda1b"),
    Formula(Equal(EisensteinG(6, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), Div(Pow(Gamma(Div(1, 3)), 18), Mul(8960, Pow(Pi, 6))))))

6c71c0

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

TeX:

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

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("6c71c0"),
    Formula(Equal(EisensteinE(6, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), Div(Mul(27, Pow(Gamma(Div(1, 3)), 18)), Mul(512, Pow(Pi, 12))))))

c6be24

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

Assumptions:

k \in \mathbb{Z}_{\ge 1}

TeX:

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

k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("c6be24"),
    Formula(Equal(EisensteinG(Mul(2, k), Mul(ConstI, Infinity)), ComplexLimit(EisensteinG(Mul(2, k), tau), For(tau, Mul(ConstI, Infinity))), Mul(2, RiemannZeta(Mul(2, k))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(1)))))

ad9ba2

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

Assumptions:

k \in \mathbb{Z}_{\ge 1}

TeX:

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

k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ad9ba2"),
    Formula(Equal(EisensteinE(Mul(2, k), Mul(ConstI, Infinity)), ComplexLimit(EisensteinE(Mul(2, k), tau), For(tau, Mul(ConstI, Infinity))), 1)),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(1)))))

e46697

\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\}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

TeX:

\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\}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane
`ModularGroupAction`	$\gamma \circ \tau$	Action of modular group
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e46697"),
    Formula(Equal(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, HH)), Set(ModularGroupAction(gamma, tau), For(Tuple(gamma, tau)), And(Element(tau, Zeros(EisensteinE(Mul(2, k), z), ForElement(z, ModularGroupFundamentalDomain))), Element(gamma, PSL2Z))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))))

2f6805

\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\}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170.

TeX:

\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\}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2f6805"),
    Formula(Subset(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(Exp(Mul(ConstI, theta)), ForElement(theta, ClosedInterval(Div(Pi, 2), Div(Mul(2, Pi), 3)))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))),
    References("F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170."))

a50278

\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170.

TeX:

\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Cardinality`	$\# S$	Set cardinality
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a50278"),
    Formula(GreaterEqual(Cardinality(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, ModularGroupFundamentalDomain))), 1)),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))),
    References("F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170."))

13cac5

\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}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19
S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf

TeX:

\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}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("13cac5"),
    Formula(Where(Equal(Sum(Mul(w(tau), ComplexZeroMultiplicity(EisensteinE(Mul(2, k), z), For(z, tau))), ForElement(tau, ModularGroupFundamentalDomain)), Div(Mul(2, k), 12)), Equal(w(tau), Cases(Tuple(Div(1, 2), Equal(tau, ConstI)), Tuple(Div(1, 3), Equal(tau, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)))), Tuple(1, Otherwise))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))),
    References("K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19", "S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf"))

097efc

\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}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19
S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf

TeX:

\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}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ModularJ`	$j(\tau)$	Modular j-invariant
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("097efc"),
    Formula(Where(Equal(Sum(Mul(Mul(ModularJ(tau), w(tau)), ComplexZeroMultiplicity(EisensteinE(Mul(2, k), z), For(z, tau))), ForElement(tau, ModularGroupFundamentalDomain)), Sub(Mul(120, k), Div(2, RiemannZeta(Sub(1, Mul(2, k)))))), Equal(w(tau), Cases(Tuple(Div(1, 2), Equal(tau, ConstI)), Tuple(1, Otherwise))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))),
    References("K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19", "S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf"))

4a200a

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{4}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{4}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("4a200a"),
    Formula(Equal(Zeros(EisensteinE(4, tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))))

ec4f56

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{6}\!\left(\tau\right) = \left\{i\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{6}\!\left(\tau\right) = \left\{i\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("ec4f56"),
    Formula(Equal(Zeros(EisensteinE(6, tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(ConstI))))

83566f

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{8}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{8}\!\left(\tau\right) = \left\{{e}^{2 \pi i / 3}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("83566f"),
    Formula(Equal(Zeros(EisensteinE(8, tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))))

26faf3

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{10}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{10}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("26faf3"),
    Formula(Equal(Zeros(EisensteinE(10, tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(ConstI, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))))

6ae250

\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}

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("6ae250"),
    Formula(Equal(Zeros(EisensteinE(12, tau), ForElement(tau, ModularGroupFundamentalDomain)), Where(Set(Div(Mul(ConstI, Hypergeometric2F1(Div(1, 6), Div(5, 6), 1, a)), Hypergeometric2F1(Div(1, 6), Div(5, 6), 1, Sub(1, a)))), Equal(a, Add(Div(1, 2), Div(Mul(Mul(21, Sqrt(10)), ConstI), 100)))))))

ad91ae

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{14}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{14}\!\left(\tau\right) = \left\{i, {e}^{2 \pi i / 3}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("ad91ae"),
    Formula(Equal(Zeros(EisensteinE(14, tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(ConstI, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))))

298bb1

\left(E_{2 k}\!\left(\tau\right) = 0 \;\mathbin{\operatorname{and}}\; \tau \notin \left\{i, {e}^{2 \pi i / 3}\right\}\right) \;\implies\; \left(\tau \notin \overline{\mathbb{Q}}\right)

Assumptions:

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathcal{F}

References:

https://doi.org/10.14992/00008713

TeX:

\left(E_{2 k}\!\left(\tau\right) = 0 \;\mathbin{\operatorname{and}}\; \tau \notin \left\{i, {e}^{2 \pi i / 3}\right\}\right) \;\implies\; \left(\tau \notin \overline{\mathbb{Q}}\right)

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathcal{F}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group

Source code for this entry:

Entry(ID("298bb1"),
    Formula(Implies(And(Equal(EisensteinE(Mul(2, k), tau), 0), NotElement(tau, Set(ConstI, Exp(Div(Mul(Mul(2, Pi), ConstI), 3))))), NotElement(tau, AlgebraicNumbers))),
    Variables(tau, k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, ModularGroupFundamentalDomain))),
    References("https://doi.org/10.14992/00008713"))

cae067

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2}\!\left(\tau\right) = \left\{ \tau + n : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{H},\,\operatorname{Re}(z) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(z\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2}\!\left(\tau\right) = \left\{ \tau + n : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{H},\,\operatorname{Re}(z) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(z\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}(z)$	Real part
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("cae067"),
    Formula(Equal(Zeros(EisensteinE(2, tau), ForElement(tau, HH)), Set(Add(tau, n), For(Tuple(tau, n)), And(Element(tau, Zeros(EisensteinE(2, z), For(z), And(Element(z, HH), Element(Re(z), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2)))))), Element(n, ZZ))))))

f33f09

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H},\,\operatorname{Re}(\tau) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(\tau\right) = \left\{ \left(\mathop{\operatorname{zero*}\,}\limits_{z \in D\left(c, d\right)} E_{2}\!\left(z\right)\right) : c \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; d \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(c, d\right) = 1 \;\mathbin{\operatorname{and}}\; -\frac{d}{c} \in \left[-\frac{1}{2}, \frac{1}{2}\right) \right\}\; \text{ where } D\!\left(c, d\right) = \operatorname{ClosedDisk}\!\left(-\frac{d}{c} + \frac{i \pi}{6 {c}^{2}}, \frac{0.000283 {\pi}^{2}}{36 {c}^{2}}\right)

References:

R. Wood and M. P. Young, Zeros of the weight two Eisenstein series, Journal of Number Theory Volume 143, October 2014, Pages 320-333. https://doi.org/10.1016/j.jnt.2014.04.007

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H},\,\operatorname{Re}(\tau) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(\tau\right) = \left\{ \left(\mathop{\operatorname{zero*}\,}\limits_{z \in D\left(c, d\right)} E_{2}\!\left(z\right)\right) : c \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; d \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(c, d\right) = 1 \;\mathbin{\operatorname{and}}\; -\frac{d}{c} \in \left[-\frac{1}{2}, \frac{1}{2}\right) \right\}\; \text{ where } D\!\left(c, d\right) = \operatorname{ClosedDisk}\!\left(-\frac{d}{c} + \frac{i \pi}{6 {c}^{2}}, \frac{0.000283 {\pi}^{2}}{36 {c}^{2}}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}(z)$	Real part
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("f33f09"),
    Formula(Equal(Zeros(EisensteinE(2, tau), ForElement(tau, HH), Element(Re(tau), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2)))), Where(Set(Parentheses(UniqueZero(EisensteinE(2, z), ForElement(z, D(c, d)))), For(Tuple(c, d)), And(Element(c, ZZ), Element(d, ZZ), Equal(GCD(c, d), 1), Element(Neg(Div(d, c)), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2))))), Equal(D(c, d), ClosedDisk(Add(Neg(Div(d, c)), Div(Mul(ConstI, Pi), Mul(6, Pow(c, 2)))), Div(Mul(Decimal("0.000283"), Pow(Pi, 2)), Mul(36, Pow(c, 2)))))))),
    References("R. Wood and M. P. Young, Zeros of the weight two Eisenstein series, Journal of Number Theory Volume 143, October 2014, Pages 320-333. https://doi.org/10.1016/j.jnt.2014.04.007"))

67f2ef

\mathop{\operatorname{zero*}\,}\limits_{y \in \left(0, \infty\right)} E_{2}\!\left(i y\right) \in \left[0.523521700017999266800534404806 \pm 1.10 \cdot 10^{-31}\right]

TeX:

\mathop{\operatorname{zero*}\,}\limits_{y \in \left(0, \infty\right)} E_{2}\!\left(i y\right) \in \left[0.523521700017999266800534404806 \pm 1.10 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("67f2ef"),
    Formula(Element(UniqueZero(EisensteinE(2, Mul(ConstI, y)), ForElement(y, OpenInterval(0, Infinity))), RealBall(Decimal("0.523521700017999266800534404806"), Decimal("1.10e-31")))))

be9790

\mathop{\operatorname{zero*}\,}\limits_{y \in \left(0, \infty\right)} E_{2}\!\left(-\frac{1}{2} + i y\right) \in \left[0.130919030396762446904114826020 \pm 2.87 \cdot 10^{-31}\right]

TeX:

\mathop{\operatorname{zero*}\,}\limits_{y \in \left(0, \infty\right)} E_{2}\!\left(-\frac{1}{2} + i y\right) \in \left[0.130919030396762446904114826020 \pm 2.87 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("be9790"),
    Formula(Element(UniqueZero(EisensteinE(2, Add(Neg(Div(1, 2)), Mul(ConstI, y))), ForElement(y, OpenInterval(0, Infinity))), RealBall(Decimal("0.130919030396762446904114826020"), Decimal("2.87e-31")))))

Definitions

Illustrations

Normalization

Lattice series

Modular transformations

Quasi-modular transformations for weight 2

Weight 2 quasi-holomorphic modular form

Fourier series (q-series)

First cases

General case

Trigonometric series

Theta function representations

Dedekind eta function representations

Elliptic function representations

Products and recurrence relations

Generating functions

Derivatives

Specific values

Fourth root of unity

Third root of unity

Infinity

Zeros

Distribution

Specific values

Transcendence

Weight 2 series