Formula:DLMF:25.15:E1

<< Lerch's Transcendent

formula in Dirichlet L-functions

Formula:DLMF:25.15:E2 >>

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle L}}}$ : Dirichlet ${\displaystyle{\displaystyle{\displaystyle L}}}$-function : http://dlmf.nist.gov/25.15#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 (1), Section 25.15 of DLMF.

URL links

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<< Lerch's Transcendent

formula in Dirichlet L-functions

Formula:DLMF:25.15:E2 >>