Fungrim entry: 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))))

Topics using this entry

Eisenstein series