Formula:DLMF:25.11:E36

<< Formula:DLMF:25.11:E35

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E37 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k}\chi(r)\HurwitzZeta@{s}{\frac{r}{k}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\chi(n)}}}$ is a Dirichlet character~${\displaystyle{\displaystyle{\displaystyle\pmod{k}}}}$

Proof

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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
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (36), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E37 >>