Fungrim entry: 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))))

Topics using this entry

Eisenstein series