DLMF:25.8.E4 (Q7661)

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DLMF:25.8.E4
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    Statements

    Defining formula
    ∑ k = 1 ∞ ( - 1 ) k k ⁢ ( ζ ⁡ ( n ⁢ k ) - 1 ) = ln ⁡ ( ∏ j = 0 n - 1 Γ ⁡ ( 2 - e ( 2 ⁢ j + 1 ) ⁢ π ⁢ i / n ) ) , superscript subscript 𝑘 1 superscript 1 𝑘 𝑘 Riemann-zeta 𝑛 𝑘 1 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 2 superscript 𝑒 2 𝑗 1 𝜋 𝑖 𝑛 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(% nk\right)-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)% \right),}}
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    DLMF id
    DLMF:25.8.E4
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    constraint
    n = 2 , 3 , 4 , … 𝑛 2 3 4 … {\displaystyle{\displaystyle n=2,3,4,\dots}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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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.m2acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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