Formula:DLMF:25.11:E11

<< Formula:DLMF:25.11:E10

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E12 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\tfrac{1}{2}}=(2^{s}-1)\RiemannZeta@{s}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

Proof

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Apply

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}}}$
and
${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}}$.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (11), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E12 >>