Formula:DLMF:25.8:E8

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k = 1 ζ ( 2 k ) k z 2 k = ln ( π z sin ( π z ) ) superscript subscript k 1 Riemann-zeta 2 k k superscript z 2 k natural-lograrithm z z {\displaystyle\sum_{k=1}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(2k% \right)}{k}z^{2k}=\mathop{\ln\/}\nolimits\!\left(\frac{\pi z}{\mathop{\sin\/}% \nolimits\!\left(\pi z\right)}\right)}


Constraint(s)


| z | < 1 𝑧 1 {\displaystyle|z|<1}


Proof


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Divide by x 𝑥 {\displaystyle x} in
k = 0 ζ ( 2 k ) z 2 k = - 1 2 π z cot ( π z ) superscript subscript k 0 Riemann-zeta 2 k superscript z 2 k 1 2 z z {\displaystyle\sum_{k=0}^{\infty}\mathop{\zeta\/}\nolimits\!\left(2k\right)z^{% 2k}=-\tfrac{1}{2}\pi z\mathop{\cot\/}\nolimits\!\left(\pi z\right)}
and integrate.



Symbols List


ζ 𝜁 {\displaystyle\zeta}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

π {\displaystyle\pi}  : the ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
cot cot {\displaystyle\mathrm{cot}}  : cotangent function : http://dlmf.nist.gov/4.14#E7
ln ln {\displaystyle\mathrm{ln}}  : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
sin {\displaystyle\sin}  : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography


Equation (6), Section 25.8 of DLMF.

URL links


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