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

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

2021-03-15 19:12:00.328586 UTC