Formula:DLMF:25.2:E9

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formula in Definition and Expansions

Formula:DLMF:25.2:E10 >>

${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}{2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{\BernoulliB{2k}}{2k}N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}}\mathrm{d}x}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>-2n}}}$

${\displaystyle{\displaystyle{\displaystyle n,N=1,2,3,\dots}}}$

Proof

Follows from

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}-1}\mathrm{d}x}}}$

by repeated integration by parts.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\widetilde{B}_{n}}}}$ : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (9), Section 25.2 of DLMF.

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