DLMF:25.16.E7 (Q7762)

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DLMF:25.16.E7
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    Defining formula
    H ⁡ ( s ) = 1 2 ⁢ ζ ⁡ ( s + 1 ) + ζ ⁡ ( s ) s - 1 - ∑ r = 1 k ( s + 2 ⁢ r - 2 2 ⁢ r - 1 ) ⁢ ζ ⁡ ( 1 - 2 ⁢ r ) ⁢ ζ ⁡ ( s + 2 ⁢ r ) - ( s + 2 ⁢ k 2 ⁢ k + 1 ) ⁢ ∑ n = 1 ∞ 1 n ⁢ ∫ n ∞ B ~ 2 ⁢ k + 1 ⁡ ( x ) x s + 2 ⁢ k + 1 ⁢ d x . Euler-sum-H 𝑠 1 2 Riemann-zeta 𝑠 1 Riemann-zeta 𝑠 𝑠 1 superscript subscript 𝑟 1 𝑘 binomial 𝑠 2 𝑟 2 2 𝑟 1 Riemann-zeta 1 2 𝑟 Riemann-zeta 𝑠 2 𝑟 binomial 𝑠 2 𝑘 2 𝑘 1 superscript subscript 𝑛 1 1 𝑛 superscript subscript 𝑛 periodic-Bernoulli-polynomial-B 2 𝑘 1 𝑥 superscript 𝑥 𝑠 2 𝑘 1 𝑥 {\displaystyle{\displaystyle H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+% \frac{\zeta\left(s\right)}{s-1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{% 2r-1}\zeta\left(1-2r\right)\zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k% }{2k+1}\sum_{n=1}^{\infty}\frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+% 1}\left(x\right)}{x^{s+2k+1}}\mathrm{d}x.}}
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    DLMF id
    DLMF:25.16.E7
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