Formula:DLMF:25.6:E6

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${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2k+1}=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoulliB{2k+1}@{t}\cot\left(\pi t\right)\mathrm{d}t}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cot}}}}$ : cotangent function : http://dlmf.nist.gov/4.14#E7
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv

Bibliography

Equation (6), Section 25.6 of DLMF.

URL links

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