Formula:DLMF:25.2:E9

<< Formula:DLMF:25.2:E8

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{B_{2k}}{2k}N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,}}$$\Riemannzeta@{s}=\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{\BernoullinumberB{2k}}{2k}N^{1-s-2k}-\binom{s+2n}{2n+1}\int_{N}^{\infty}\frac{\perBernoulliB{2n+1}@{x}}{x^{s+2n+1}}\diff{x},$

Constraint(s)

${\displaystyle{\displaystyle\Re s>-2n}}$$\Re s>-2n$
${\displaystyle{\displaystyle n,N=1,2,3,\dots}}$$n,N=1,2,3,\dots$

Proof

Follows from

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

by repeated integration by parts.

Symbols List

• ${\displaystyle{\displaystyle B_{\NVar{n}}}}$$\BernoullinumberB{\NVar{n}}$: Q11109
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$$\Riemannzeta@{\NVar{s}}$: Q10579
• ${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$$\binom{\NVar{m}}{\NVar{n}}$: Q10754
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$$\diff{\NVar{x}}$: Q10770
• ${\displaystyle{\displaystyle\int}}$$\int$: Q10771
• ${\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}$$\perBernoulliB{\NVar{n}}@{\NVar{x}}$: Q11110
• ${\displaystyle{\displaystyle\Re}}$$\realpart@@$: Q10811
• ${\displaystyle{\displaystyle k}}$$k$: Q12150
• ${\displaystyle{\displaystyle n}}$$n$: Q12148
• ${\displaystyle{\displaystyle x}}$$x$: Q12152
• ${\displaystyle{\displaystyle s}}$$s$: Q12149

& : 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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