DLMF:25.11.E16 (Q7690)

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DLMF:25.11.E16
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    Statements

    Defining formula
    ζ ⁡ ( 1 - s , h k ) = 2 ⁢ Γ ⁡ ( s ) ( 2 ⁢ π ⁢ k ) s ⁢ ∑ r = 1 k cos ⁡ ( π ⁢ s 2 - 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ζ ⁡ ( s , r k ) , Hurwitz-zeta 1 𝑠 ℎ 𝑘 2 Euler-Gamma 𝑠 superscript 2 𝜋 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 𝜋 𝑠 2 2 𝜋 𝑟 ℎ 𝑘 Hurwitz-zeta 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma% \left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2% \pi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right),}}
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    DLMF id
    DLMF:25.11.E16
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    constraint
    1 ≤ h ≤ k 1 ℎ 𝑘 {\displaystyle{\displaystyle 1\leq h\leq k}}
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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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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aqdec
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