Formula:DLMF:25.15:E3

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formula in Dirichlet L-functions

Formula:DLMF:25.15:E4 >>

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right)}}}$

Constraint(s)

hold for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ if ${\displaystyle{\displaystyle{\displaystyle\chi\neq\chi_{1}}}}$, and for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ (${\displaystyle{\displaystyle{\displaystyle\neq 1}}}$) if ${\displaystyle{\displaystyle{\displaystyle\chi=\chi_{1}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle L}}}$ : Dirichlet ${\displaystyle{\displaystyle{\displaystyle L}}}$-function : http://dlmf.nist.gov/25.15#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \zeta}}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1

Bibliography

Equation (3), Section 25.15 of DLMF.

URL links

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