DLMF:25.2.E8 (Q7603)

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DLMF:25.2.E8
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    Statements

    Defining formula
    ζ ⁡ ( s ) = ∑ k = 1 N 1 k s + N 1 - s s - 1 - s ⁢ ∫ N ∞ x - ⌊ x ⌋ x s + 1 ⁢ d x , Riemann-zeta 𝑠 superscript subscript 𝑘 1 𝑁 1 superscript 𝑘 𝑠 superscript 𝑁 1 𝑠 𝑠 1 𝑠 superscript subscript 𝑁 𝑥 𝑥 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+% \frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{% s+1}}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.2.E8
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    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    N = 1 , 2 , 3 , … 𝑁 1 2 3 … {\displaystyle{\displaystyle N=1,2,3,\dots}}
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