Reflection Formulas

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Reflection Formulas

\RiemannZeta ⁢ @ ⁢ 1 - s = 2 ⁢ ( 2 ⁢ π ) - s ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s ) ⁢ Γ ⁡ ( s ) ⁢ \RiemannZeta ⁢ @ ⁢ s \RiemannZeta @ 1 𝑠 2 superscript 2 𝑠 1 2 𝑠 Euler-Gamma 𝑠 \RiemannZeta @ 𝑠 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{1-s}=2(2\pi)^{-s}\cos% \left(\tfrac{1}{2}\pi s\right)\Gamma\left(s\right)\RiemannZeta@{s}}}} {\displaystyle \RiemannZeta@{1-s} = 2 (2\cpi)^{-s} \cos@{\tfrac{1}{2} \cpi s} \EulerGamma@{s} \RiemannZeta@{s} }

Constraint(s): s ≠ 0 , 1 𝑠 0 1 {\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}


\RiemannZeta ⁢ @ ⁢ s = 2 ⁢ ( 2 ⁢ π ) s - 1 ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ s ) ⁢ Γ ⁡ ( 1 - s ) ⁢ \RiemannZeta ⁢ @ ⁢ 1 - s \RiemannZeta @ 𝑠 2 superscript 2 𝑠 1 1 2 𝑠 Euler-Gamma 1 𝑠 \RiemannZeta @ 1 𝑠 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=2(2\pi)^{s-1}\sin% \left(\tfrac{1}{2}\pi s\right)\Gamma\left(1-s\right)\RiemannZeta@{1-s}}}} {\displaystyle \RiemannZeta@{s} = 2 (2\cpi)^{s-1} \sin@{\tfrac{1}{2} \cpi s} \EulerGamma@{1-s} \RiemannZeta@{1-s} }
\RiemannXi ⁢ @ ⁢ s = \RiemannXi ⁢ @ ⁢ 1 - s \RiemannXi @ 𝑠 \RiemannXi @ 1 𝑠 {\displaystyle{\displaystyle{\displaystyle\RiemannXi@{s}=\RiemannXi@{1-s}}}} {\displaystyle \RiemannXi@{s} = \RiemannXi@{1-s} }
\RiemannXi ⁢ @ ⁢ s = 1 2 ⁢ s ⁢ ( s - 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ π - s / 2 ⁢ \RiemannZeta ⁢ @ ⁢ s \RiemannXi @ 𝑠 1 2 𝑠 𝑠 1 Euler-Gamma 1 2 𝑠 𝑠 2 \RiemannZeta @ 𝑠 {\displaystyle{\displaystyle{\displaystyle\RiemannXi@{s}=\tfrac{1}{2}s(s-1)% \Gamma\left(\tfrac{1}{2}s\right){\pi^{-s/2}}\RiemannZeta@{s}}}} {\displaystyle \RiemannXi@{s} = \tfrac{1}{2} s (s-1) \EulerGamma@{\tfrac{1}{2} s} \cpi^{-s/2} \RiemannZeta@{s} }
( - 1 ) k ⁢ \RiemannZeta ( k ) ⁢ @ ⁢ 1 - s = 2 ( 2 ⁢ π ) s ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ( ℜ ⁡ ( c k - m ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s ) + ℑ ⁡ ( c k - m ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ s ) ) ⁢ Γ ( r ) ⁡ ( s ) ⁢ \RiemannZeta ( m - r ) ⁢ @ ⁢ s superscript 1 𝑘 superscript \RiemannZeta 𝑘 @ 1 𝑠 2 superscript 2 𝑠 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 1 2 𝑠 superscript 𝑐 𝑘 𝑚 1 2 𝑠 Euler-Gamma 𝑟 𝑠 superscript \RiemannZeta 𝑚 𝑟 @ 𝑠 {\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{1-s}=% \frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re{(c^{k-m})}\cos\left(\tfrac{1}{2}\pi s% \right)+\Im{(c^{k-m})}\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}% \left(s\right)\RiemannZeta^{(m-r)}@{s}}}} {\displaystyle \opminus^k \RiemannZeta^{(k)}@{1-s} = \frac{2}{(2\cpi)^s} \sum_{m=0}^k \sum_{r=0}^m \binom{k}{m} \binom{m}{r} \left( \realpart{(c^{k-m})} \cos@{\tfrac{1}{2} \cpi s} + \imagpart{(c^{k-m})} \sin@{\tfrac{1}{2} \cpi s} \right) \EulerGamma^{(r)}@{s} \RiemannZeta^{(m-r)}@{s} }

Substitution(s): c = - ln ⁡ ( 2 ⁢ π ) - 1 2 ⁢ π ⁢ i 𝑐 2 1 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle{\displaystyle c=-\ln\left(2\pi% \right)-\tfrac{1}{2}\pi\mathrm{i}}}}}


Constraint(s): s ≠ 0 , 1 𝑠 0 1 {\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}} &
k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}


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