Formula:DLMF:25.8:E7

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Formula:DLMF:25.8:E8 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\RiemannZeta@{k}}{k}z^{k}=-\gamma z+\ln\Gamma\left(1-z\right)}}}$

Constraint(s)

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

Proof

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

${\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\RiemannZeta@{k}z^{k}=-\gamma z-z\psi\left(1-z\right)}}}$

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
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ : Euler's constant : http://dlmf.nist.gov/5.2#E3
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation (7), Section 25.8 of DLMF.

URL links

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