Formula:DLMF:25.11:E24

<< Formula:DLMF:25.11:E23

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E25 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{k-1}\HurwitzZeta^{\prime}@{s}{\frac{r}{k}}=(k^{s}-1)\RiemannZeta^{\prime}@{s}+k^{s}\RiemannZeta@{s}\ln k}}}$

Constraint(s)

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

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

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Use

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{ka}=k^{-s}\*\sum_{n=0}^{k-1}\HurwitzZeta@{s}{a+\frac{n}{k}}}}}$
with ${\displaystyle{\displaystyle{\displaystyle a=1/k}}}$,

multiply by ${\displaystyle{\displaystyle{\displaystyle k^{s}}}}$ and differentiate.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : 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

Bibliography

Equation (24), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E23

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E25 >>