Formula:DLMF:25.11:E11: Difference between revisions

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Latest revision as of 08:32, 22 December 2019


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

Constraint(s)

s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


Proof

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Apply

\RiemannZeta ⁢ @ ⁢ s = 1 1 - 2 - s ⁢ ∑ n = 0 ∞ 1 ( 2 ⁢ n + 1 ) s \RiemannZeta @ 𝑠 1 1 superscript 2 𝑠 superscript subscript 𝑛 0 1 superscript 2 𝑛 1 𝑠 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{1-2^{-s}}% \sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{1 - 2^{-s}} \sum_{n=0}^\infty \frac{1}{(2n+1)^s} }
and
\HurwitzZeta ⁢ @ ⁢ s ⁢ a = ∑ n = 0 ∞ 1 ( n + a ) s \HurwitzZeta @ 𝑠 𝑎 superscript subscript 𝑛 0 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{% \infty}\frac{1}{(n+a)^{s}}}}} {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{1}{(n+a)^s} } .


Symbols List

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