DLMF:25.11.E37 (Q7711)

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DLMF:25.11.E37
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    Statements

    test
    ∑ 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),}}
    0 references
    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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    n = 2 , 3 , 4 , … 𝑛 2 3 4 … {\displaystyle{\displaystyle n=2,3,4,\dots}}
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    ℜ ⁡ a ≥ 1 𝑎 1 {\displaystyle{\displaystyle\Re a\geq 1}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ajdec
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    Hurwitz zeta function
    test
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aakdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2agdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1atdec
    0 references
     
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