\RiemannZeta @ 0 = - 1 2 \RiemannZeta @ 0 1 2 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{0}=-\frac{1}{2}}}} {\displaystyle \RiemannZeta@{0} = - \frac{1}{2} } \RiemannZeta @ 2 = π 2 6 \RiemannZeta @ 2 2 6 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2}=\frac{{\pi^{2}}}{6}% }}} {\displaystyle \RiemannZeta@{2} = \frac{\cpi^2}{6} } \RiemannZeta @ 4 = π 4 90 \RiemannZeta @ 4 4 90 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{4}=\frac{{\pi^{4}}}{90% }}}} {\displaystyle \RiemannZeta@{4} = \frac{\cpi^4}{90} } \RiemannZeta @ 6 = π 6 945 \RiemannZeta @ 6 6 945 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{6}=\frac{{\pi^{6}}}{94% 5}}}} {\displaystyle \RiemannZeta@{6} = \frac{\cpi^6}{945} } \RiemannZeta @ 2 n = ( 2 π ) 2 n 2 ( 2 n ) ! | \BernoulliB 2 n | \RiemannZeta @ 2 𝑛 superscript 2 2 𝑛 2 2 𝑛 \BernoulliB 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2n}=\frac{(2\pi)^{2n}}% {2(2n)!}\left|\BernoulliB{2n}\right|}}} {\displaystyle \RiemannZeta@{2n} = \frac{(2\cpi)^{2n}}{2(2n)!} \left| \BernoulliB{2n} \right| }
\RiemannZeta @ - n = - \BernoulliB n + 1 n + 1 \RiemannZeta @ 𝑛 \BernoulliB 𝑛 1 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{-n}=-\frac{\BernoulliB% {n+1}}{n+1}}}} {\displaystyle \RiemannZeta@{-n} = -\frac{\BernoulliB{n+1}}{n+1} }
\RiemannZeta @ - 2 n = 0 \RiemannZeta @ 2 𝑛 0 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{-2n}=0}}} {\displaystyle \RiemannZeta@{-2n} = 0 }
\RiemannZeta @ k + 1 = 1 k ! ∑ n 1 = 1 ∞ … ∑ n k = 1 ∞ 1 n 1 ⋯ n k ( n 1 + … + n k ) \RiemannZeta @ 𝑘 1 1 𝑘 superscript subscript subscript 𝑛 1 1 … superscript subscript subscript 𝑛 𝑘 1 1 subscript 𝑛 1 ⋯ subscript 𝑛 𝑘 subscript 𝑛 1 … subscript 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{k+1}=\frac{1}{k!}\sum_% {n_{1}=1}^{\infty}\dots\sum_{n_{k}=1}^{\infty}\frac{1}{n_{1}\cdots n_{k}(n_{1}% +\dots+n_{k})}}}} {\displaystyle \RiemannZeta@{k+1} = \frac{1}{k!} \sum_{n_1=1}^\infty \dots \sum_{n_k=1}^\infty \frac{1}{n_1 \cdots n_k (n_1 + \dots + n_k)} }
\RiemannZeta @ 2 k + 1 = ( - 1 ) k + 1 ( 2 π ) 2 k + 1 2 ( 2 k + 1 ) ! ∫ 0 1 \BernoulliB 2 k + 1 @ t cot ( π t ) d t \RiemannZeta @ 2 𝑘 1 superscript 1 𝑘 1 superscript 2 2 𝑘 1 2 2 𝑘 1 superscript subscript 0 1 \BernoulliB 2 𝑘 1 @ 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2k+1}=\frac{(-1)^{k+1}% (2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoulliB{2k+1}@{t}\cot\left(\pi t\right% )\mathrm{d}t}}} {\displaystyle \RiemannZeta@{2k+1} = \frac{\opminus^{k+1} (2 \cpi)^{2k+1}}{2(2k+1)!} \int_0^1 \BernoulliB{2k+1}@{t} \cot@{\cpi t} \diff{t} }
\RiemannZeta @ 2 = ∫ 0 1 ∫ 0 1 1 1 - x y d x d y \RiemannZeta @ 2 superscript subscript 0 1 superscript subscript 0 1 1 1 𝑥 𝑦 𝑥 𝑦 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2}=\int_{0}^{1}\int_{0% }^{1}\frac{1}{1-xy}\mathrm{d}x\mathrm{d}y}}} {\displaystyle \RiemannZeta@{2} = \int_0^1 \int_0^1 \frac{1}{1-xy} \diff{x} \diff{y} } \RiemannZeta @ 2 = 3 ∑ k = 1 ∞ 1 k 2 ( 2 k k ) \RiemannZeta @ 2 3 superscript subscript 𝑘 1 1 superscript 𝑘 2 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{2}=3\sum_{k=1}^{\infty% }\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}} {\displaystyle \RiemannZeta@{2} = 3 \sum_{k=1}^\infty \frac{1}{k^2 \binom{2k}{k}} } \RiemannZeta @ 3 = 5 2 ∑ k = 1 ∞ ( - 1 ) k - 1 k 3 ( 2 k k ) \RiemannZeta @ 3 5 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 superscript 𝑘 3 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{3}=\frac{5}{2}\sum_{k=% 1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}} {\displaystyle \RiemannZeta@{3} = \frac{5}{2} \sum_{k=1}^\infty \frac{\opminus^{k-1}}{k^3 \binom{2k}{k}} } \RiemannZeta @ 4 = 36 17 ∑ k = 1 ∞ 1 k 4 ( 2 k k ) \RiemannZeta @ 4 36 17 superscript subscript 𝑘 1 1 superscript 𝑘 4 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{4}=\frac{36}{17}\sum_{% k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}} {\displaystyle \RiemannZeta@{4} = \frac{36}{17} \sum_{k=1}^\infty \frac{1}{k^4 \binom{2k}{k}} }
\RiemannZeta ′ @ 0 = - 1 2 ln ( 2 π ) superscript \RiemannZeta ′ @ 0 1 2 2 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta^{\prime}@{0}=-\tfrac{1}% {2}\ln\left(2\pi\right)}}} {\displaystyle \RiemannZeta'@{0} = - \tfrac{1}{2} \ln@{2\cpi} } \RiemannZeta ′′ @ 0 = - 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 - 1 24 π 2 + γ 1 superscript \RiemannZeta ′′ @ 0 1 2 superscript 2 2 1 2 2 1 24 2 Stieltjes-constants 1 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta^{\prime\prime}@{0}=-% \tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}{% \pi^{2}}+\gamma_{1}}}} {\displaystyle \RiemannZeta''@{0} = -\tfrac{1}{2} (\ln@{2\cpi})^2 + \tfrac{1}{2} \EulerConstant^2 - \tfrac{1}{24} \cpi^2 + \StieltjesConstants{1} } ( - 1 ) k \RiemannZeta ( k ) @ - 2 n = 2 ( - 1 ) n ( 2 π ) 2 n + 1 ∑ m = 0 k ∑ r = 0 m ( k m ) ( m r ) ℑ ( c k - m ) Γ ( r ) ( 2 n + 1 ) \RiemannZeta ( m - r ) @ 2 n + 1 superscript 1 𝑘 superscript \RiemannZeta 𝑘 @ 2 𝑛 2 superscript 1 𝑛 superscript 2 2 𝑛 1 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 Euler-Gamma 𝑟 2 𝑛 1 superscript \RiemannZeta 𝑚 𝑟 @ 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{-2n}=% \frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0% pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im{(c^{k-m})}\*{\Gamma^{(r)}}\left(2n% +1\right)\RiemannZeta^{(m-r)}@{2n+1}}}} {\displaystyle \opminus^k \RiemannZeta^{(k)}@{-2n} = \frac{2 \opminus^n}{(2\cpi)^{2n+1}} \sum_{m=0}^k \sum_{r=0}^m \binom{k}{m} \binom{m}{r} \imagpart{(c^{k-m})} \* \EulerGamma^{(r)}@{2n+1} \RiemannZeta^{(m-r)}@{2n+1} }
( - 1 ) k \RiemannZeta ( k ) @ 1 - 2 n = 2 ( - 1 ) n ( 2 π ) 2 n ∑ m = 0 k ∑ r = 0 m ( k m ) ( m r ) ℜ ( c k - m ) Γ ( r ) ( 2 n ) \RiemannZeta ( m - r ) @ 2 n superscript 1 𝑘 superscript \RiemannZeta 𝑘 @ 1 2 𝑛 2 superscript 1 𝑛 superscript 2 2 𝑛 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 Euler-Gamma 𝑟 2 𝑛 superscript \RiemannZeta 𝑚 𝑟 @ 2 𝑛 {\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{1-2n}=% \frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}% {}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Re{(c^{k-m})}\*{\Gamma^{(r)}}\left(2n% \right)\RiemannZeta^{(m-r)}@{2n}}}} {\displaystyle \opminus^k \RiemannZeta^{(k)}@{1-2n} = \frac{2 \opminus^n}{(2\cpi)^{2n}} \sum_{m=0}^k \sum_{r=0}^m \binom{k}{m} \binom{m}{r} \realpart{(c^{k-m})} \* \EulerGamma^{(r)}@{2n} \RiemannZeta^{(m-r)}@{2n} }
\RiemannZeta ′ @ 2 n = ( - 1 ) n + 1 ( 2 π ) 2 n 2 ( 2 n ) ! ( 2 n \RiemannZeta ′ @ 1 - 2 n - ( ψ ( 2 n ) - ln ( 2 π ) ) \BernoulliB 2 n ) superscript \RiemannZeta ′ @ 2 𝑛 superscript 1 𝑛 1 superscript 2 2 𝑛 2 2 𝑛 2 𝑛 superscript \RiemannZeta ′ @ 1 2 𝑛 digamma 2 𝑛 2 \BernoulliB 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta^{\prime}@{2n}=\frac{(-1% )^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\RiemannZeta^{\prime}@{1-2n}-(\psi\left(2n% \right)-\ln\left(2\pi\right))\BernoulliB{2n}\right)}}} {\displaystyle \RiemannZeta'@{2n} = \frac{\opminus^{n+1} (2\cpi)^{2n}}{2(2n)!} \left( 2n \RiemannZeta'@{1-2n} - (\digamma@{2n} - \ln@{2\cpi}) \BernoulliB{2n} \right) }
( n + 1 2 ) \RiemannZeta @ 2 n = ∑ k = 1 n - 1 \RiemannZeta @ 2 k \RiemannZeta @ 2 n - 2 k 𝑛 1 2 \RiemannZeta @ 2 𝑛 superscript subscript 𝑘 1 𝑛 1 \RiemannZeta @ 2 𝑘 \RiemannZeta @ 2 𝑛 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(n+\tfrac{1}{2}\right)% \RiemannZeta@{2n}=\sum_{k=1}^{n-1}\RiemannZeta@{2k}\RiemannZeta@{2n-2k}}}} {\displaystyle \left( n + \tfrac{1}{2} \right) \RiemannZeta@{2n} = \sum_{k=1}^{n-1} \RiemannZeta@{2k} \RiemannZeta@{2n-2k} }
( n + 3 4 ) \RiemannZeta @ 4 n + 2 = ∑ k = 1 n \RiemannZeta @ 2 k \RiemannZeta @ 4 n + 2 - 2 k 𝑛 3 4 \RiemannZeta @ 4 𝑛 2 superscript subscript 𝑘 1 𝑛 \RiemannZeta @ 2 𝑘 \RiemannZeta @ 4 𝑛 2 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(n+\tfrac{3}{4}\right)% \RiemannZeta@{4n+2}=\sum_{k=1}^{n}\RiemannZeta@{2k}\RiemannZeta@{4n+2-2k}}}} {\displaystyle \left( n + \tfrac{3}{4} \right) \RiemannZeta@{4n+2} = \sum_{k=1}^n \RiemannZeta@{2k} \RiemannZeta@{4n+2-2k} }
( n + 1 4 ) \RiemannZeta @ 4 n + 1 2 ( \RiemannZeta @ 2 n ) 2 = ∑ k = 1 n \RiemannZeta @ 2 k \RiemannZeta @ 4 n - 2 k 𝑛 1 4 \RiemannZeta @ 4 𝑛 1 2 superscript \RiemannZeta @ 2 𝑛 2 superscript subscript 𝑘 1 𝑛 \RiemannZeta @ 2 𝑘 \RiemannZeta @ 4 𝑛 2 𝑘 {\displaystyle{\displaystyle{\displaystyle{\left(n+\tfrac{1}{4}\right)% \RiemannZeta@{4n}+\tfrac{1}{2}(\RiemannZeta@{2n})^{2}=\sum_{k=1}^{n}% \RiemannZeta@{2k}\RiemannZeta@{4n-2k}}}}} {\displaystyle {\left( n + \tfrac{1}{4} \right) \RiemannZeta@{4n} + \tfrac{1}{2} (\RiemannZeta@{2n})^2 = \sum_{k=1}^n \RiemannZeta@{2k} \RiemannZeta@{4n-2k}} }
( m + n + 3 2 ) \RiemannZeta @ 2 m + 2 n + 2 = ( ∑ k = 1 m + ∑ k = 1 n ) \RiemannZeta @ 2 k \RiemannZeta @ 2 m + 2 n + 2 - 2 k 𝑚 𝑛 3 2 \RiemannZeta @ 2 𝑚 2 𝑛 2 superscript subscript 𝑘 1 𝑚 superscript subscript 𝑘 1 𝑛 \RiemannZeta @ 2 𝑘 \RiemannZeta @ 2 𝑚 2 𝑛 2 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(m+n+\tfrac{3}{2}\right)% \RiemannZeta@{2m+2n+2}=\left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\RiemannZeta@% {2k}\RiemannZeta@{2m+2n+2-2k}}}} {\displaystyle \left( m + n + \tfrac{3}{2} \right) \RiemannZeta@{2m+2n+2} = \left( \sum_{k=1}^m + \sum_{k=1}^n \right) \RiemannZeta@{2k} \RiemannZeta@{2m+2n+2-2k} }
1 2 ( 2 2 n - 1 ) \RiemannZeta @ 2 n = ∑ k = 1 n - 1 ( 2 2 n - 2 k - 1 ) \RiemannZeta @ 2 n - 2 k \RiemannZeta @ 2 k 1 2 superscript 2 2 𝑛 1 \RiemannZeta @ 2 𝑛 superscript subscript 𝑘 1 𝑛 1 superscript 2 2 𝑛 2 𝑘 1 \RiemannZeta @ 2 𝑛 2 𝑘 \RiemannZeta @ 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\tfrac{1}{2}(2^{2n}-1)\RiemannZeta@{% 2n}=\sum_{k=1}^{n-1}(2^{2n-2k}-1)\RiemannZeta@{2n-2k}\RiemannZeta@{2k}}}} {\displaystyle \tfrac{1}{2} (2^{2n} - 1) \RiemannZeta@{2n} = \sum_{k=1}^{n-1} (2^{2n-2k} - 1) \RiemannZeta@{2n-2k} \RiemannZeta@{2k} }