# Fungrim entry: 72ccda

$\zeta\!\left(2 n\right) = \frac{{\left(-1\right)}^{n + 1} B_{2 n} {\left(2 \pi\right)}^{2 n}}{2 \left(2 n\right)!}$
Assumptions:$n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ge 1$
TeX:
\zeta\!\left(2 n\right) = \frac{{\left(-1\right)}^{n + 1} B_{2 n} {\left(2 \pi\right)}^{2 n}}{2 \left(2 n\right)!}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ge 1
Definitions:
Fungrim symbol Notation Short description
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Pow${a}^{b}$ Power
BernoulliB$B_{n}$ Bernoulli number
Pi$\pi$ The constant pi (3.14...)
Factorial$n !$ Factorial
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("72ccda"),
Formula(Equal(RiemannZeta(Mul(2, n)), Div(Mul(Mul(Pow(-1, Add(n, 1)), BernoulliB(Mul(2, n))), Pow(Mul(2, Pi), Mul(2, n))), Mul(2, Factorial(Mul(2, n)))))),
Variables(n),
Assumptions(And(Element(n, ZZ), GreaterEqual(n, 1))))

## 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