Formula:DLMF:25.16:E7

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Formula:DLMF:25.16:E8:SE1 >>

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}=\frac{1}{2}\RiemannZeta@{s+1}+\frac{\RiemannZeta@{s}}{s-1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\RiemannZeta@{1-2r}\RiemannZeta@{s+2r}-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}\frac{1}{n}\int_{n}^{\infty}\frac{\PeriodicBernoulliB{2k+1}@{x}}{x^{s+2k+1}}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>-2k}}}$ for every positive integer ${\displaystyle{\displaystyle{\displaystyle k}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle H}}}$ : Euler sums : http://dlmf.nist.gov/25.16#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : 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 (7), Section 25.16 of DLMF.

URL links

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