Formula:DLMF:25.8:E8

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${\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\RiemannZeta@{2k}}{k}z^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle|z|<1}}}$

Proof

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Divide by ${\displaystyle{\displaystyle{\displaystyle x}}}$ in

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\RiemannZeta@{2k}z^{2k}=-\tfrac{1}{2}\pi z\cot\left(\pi z\right)}}}$

and integrate.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation (8), Section 25.8 of DLMF.

URL links

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