Fungrim entry: 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))))

Topics using this entry

Eisenstein series