DLMF:25.11.E37 (Q7711)

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DLMF:25.11.E37
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    Statements

    Defining formula
    ∑ k = 1 ∞ ( - 1 ) k k ⁢ ζ ⁡ ( n ⁢ k , a ) = - n ⁢ ln ⁡ Γ ⁡ ( a ) + ln ⁡ ( ∏ j = 0 n - 1 Γ ⁡ ( a - e ( 2 ⁢ j + 1 ) ⁢ π ⁢ i / n ) ) , superscript subscript 𝑘 1 superscript 1 𝑘 𝑘 Hurwitz-zeta 𝑛 𝑘 𝑎 𝑛 Euler-Gamma 𝑎 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 𝑎 superscript 𝑒 2 𝑗 1 𝜋 𝑖 𝑛 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk% ,a\right)=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^% {(2j+1)\pi i/n}\right)\right),}}
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    DLMF id
    DLMF:25.11.E37
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    constraint
    ℜ ⁡ a ≥ 1 𝑎 1 {\displaystyle{\displaystyle\Re a\geq 1}}
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