Formula:DLMF:25.11:E11

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ζ ( s , 1 2 ) = ( 2 s - 1 ) ζ ( s ) Hurwitz-zeta s 1 2 superscript 2 s 1 Riemann-zeta s {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}\right)=(2^{s}-1)% \mathop{\zeta\/}\nolimits\!\left(s\right)}


Constraint(s)


Failed to parse(LaTeXML Invalid response ('Error fetching URL: name lookup timed out') from server 'http://latexml.mathweb.org/convert':): {\displaystyle s \neq 1}


Proof


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ζ ( s ) = 1 1 - 2 - s n = 0 1 ( 2 n + 1 ) s Riemann-zeta s 1 1 superscript 2 s superscript subscript n 0 1 superscript 2 n 1 s {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{1-2^{-s}}\sum% _{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}

and

ζ ( s , a ) = n = 0 1 ( n + a ) s Hurwitz-zeta s a superscript subscript n 0 1 superscript n a s {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{\infty}% \frac{1}{(n+a)^{s}}} .



Symbols List


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

Bibliography


Equation (1), Section 25.11 of DLMF.

URL links


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