# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC