DLMF:25.2.E9 (Q7604)

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DLMF:25.2.E9
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    Statements

    Defining formula
    ζ ⁡ ( s ) = ∑ k = 1 N 1 k s + N 1 - s s - 1 - 1 2 ⁢ N - s + ∑ k = 1 n ( s + 2 ⁢ k - 2 2 ⁢ k - 1 ) ⁢ B 2 ⁢ k 2 ⁢ k ⁢ N 1 - s - 2 ⁢ k - ( s + 2 ⁢ n 2 ⁢ n + 1 ) ⁢ ∫ N ∞ B ~ 2 ⁢ n + 1 ⁡ ( x ) x s + 2 ⁢ n + 1 ⁢ d x , Riemann-zeta 𝑠 superscript subscript 𝑘 1 𝑁 1 superscript 𝑘 𝑠 superscript 𝑁 1 𝑠 𝑠 1 1 2 superscript 𝑁 𝑠 superscript subscript 𝑘 1 𝑛 binomial 𝑠 2 𝑘 2 2 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 superscript 𝑁 1 𝑠 2 𝑘 binomial 𝑠 2 𝑛 2 𝑛 1 superscript subscript 𝑁 periodic-Bernoulli-polynomial-B 2 𝑛 1 𝑥 superscript 𝑥 𝑠 2 𝑛 1 𝑥 {\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,}}
    0 references
    DLMF id
    DLMF:25.2.E9
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    constraint
    ℜ ⁡ s > - 2 ⁢ n 𝑠 2 𝑛 {\displaystyle{\displaystyle\Re s>-2n}}
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    n , N = 1 , 2 , 3 , … formulae-sequence 𝑛 𝑁 1 2 3 … {\displaystyle{\displaystyle n,N=1,2,3,\dots}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2agdec
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    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    Q11110
    Defining formula
    B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS3.m1adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
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    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1cdec
    0 references
    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1gdec
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    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1adec
    0 references
    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1gdec
    0 references
     
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