DLMF:25.16.E6 (Q7761)

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DLMF:25.16.E6
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    Statements

    Defining formula
    H ⁡ ( s ) = - ζ ′ ⁡ ( s ) + γ ⁢ ζ ⁡ ( s ) + 1 2 ⁢ ζ ⁡ ( s + 1 ) + ∑ r = 1 k ζ ⁡ ( 1 - 2 ⁢ r ) ⁢ ζ ⁡ ( s + 2 ⁢ r ) + ∑ n = 1 ∞ 1 n s ⁢ ∫ n ∞ B ~ 2 ⁢ k + 1 ⁡ ( x ) x 2 ⁢ k + 2 ⁢ d x , Euler-sum-H 𝑠 diffop Riemann-zeta 1 𝑠 Riemann-zeta 𝑠 1 2 Riemann-zeta 𝑠 1 superscript subscript 𝑟 1 𝑘 Riemann-zeta 1 2 𝑟 Riemann-zeta 𝑠 2 𝑟 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 superscript subscript 𝑛 periodic-Bernoulli-polynomial-B 2 𝑘 1 𝑥 superscript 𝑥 2 𝑘 2 𝑥 {\displaystyle{\displaystyle H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta% \left(s\right)+\frac{1}{2}\zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r% \right)\zeta\left(s+2r\right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{% \infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{2k+2}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.16.E6
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    Q12169
    Defining formula
    H ⁡ ( s ) Euler-sum-H 𝑠 {\displaystyle{\displaystyle H\left(\NVar{s}\right)}}
    xml-id
    C25.S16.SS2.p1.m1aadec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aadec
    0 references
     
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