Fungrim entry: 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))))

Topics using this entry

Eisenstein series