Formula:DLMF:25.11:E15

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E16 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

Proof

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Use

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}}$

and analytic continuation.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (15), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E16 >>