Formula:DLMF:25.2:E9

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ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - 1 2 N - s + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k N 1 - s - 2 k - ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 x Riemann-zeta s superscript subscript k 1 N 1 superscript k s superscript N 1 s s 1 1 2 superscript N s superscript subscript k 1 n binomial s 2 k 2 2 k 1 Bernoulli-polynomial-B 2 k 2 k superscript N 1 s 2 k binomial s 2 n 2 n 1 superscript subscript N periodic-Bernoulli-polynomial-B 2 n 1 x superscript x s 2 n 1 x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum_{k=1}^{N}\frac{1}% {k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}{2}N^{-s}+\sum_{k=1}^{n}\binom{s+2k-2}{2k-% 1}\frac{\mathop{B_{2k}\/}\nolimits}{2k}N^{1-s-2k}-\binom{s+2n}{2n+1}\int_{N}^{% \infty}\frac{\mathop{\widetilde{B}_{2n+1}\/}\nolimits\!\left(x\right)}{x^{s+2n% +1}}dx}


Constraint(s)


Failed to parse(LaTeXML Invalid response ('Error fetching URL: name lookup timed out') from server 'http://latexml.mathweb.org/convert':): {\displaystyle \realpart{s} > -2n} & n , N = 1 , 2 , 3 , formulae-sequence 𝑛 𝑁 1 2 3 {\displaystyle n,N=1,2,3,\dots}


Proof


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Follows from
ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - s N x - x x s + 1 x Riemann-zeta s superscript subscript k 1 N 1 superscript k s superscript N 1 s s 1 s superscript subscript N x x superscript x s 1 x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum_{k=1}^{N}\frac{1}% {k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right% \rfloor}{x^{s+1}}dx}
by repeated integration by parts.



Symbols List


ζ 𝜁 {\displaystyle\zeta}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
{\displaystyle\int}  : integral : http://dlmf.nist.gov/1.4#iv
a 𝑎 {\displaystyle\left\lfloor\hskip 0.0pta\right\rfloor}  : floor : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16
a b 𝑏 𝑎 {\displaystyle\diff[a]{b}}  : differential : http://dlmf.nist.gov/1.4#iv
( a b ) binomial 𝑎 𝑏 {\displaystyle\binom{\hskip 0.0pta}{\hskip 0.0ptb}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
B a subscript 𝐵 𝑎 {\displaystyle B_{\hskip 0.0pta}}  : Bernoulli polynomials : http://dlmf.nist.gov/24.2#i
B ~ a subscript ~ 𝐵 𝑎 {\displaystyle\widetilde{B}_{\hskip 0.0pta}}  : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
a 𝑎 {\displaystyle\Re{\,\hskip 0.0pta}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography


Equation (8), Section 25.2 of DLMF.

URL links


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