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Patterns in zeta function

  1. May 13, 2006 #1
    I'm experimenting with zeta function right now, and I assume there must be some kind of patter in zetas of consecutive (even) numbers.

    For example when we do,
    [tex]\zeta(2)=\pi^2 /6 [/tex]

    [tex]\zeta(4)=\pi^4/90 [/tex]

    [tex]\zeta(6)=\pi^6/945 [/tex]

    [tex]\zeta(8)=\pi^8/9450 [/tex]

    However,

    [tex]\zeta(12)=691\pi^{12}/638512875 [/tex]

    So, Can anyone explain this pattern to me?
    I'd really appreciate
     
    Last edited: May 13, 2006
  2. jcsd
  3. May 13, 2006 #2

    Curious3141

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  4. May 14, 2006 #3
    Thanks Curious,
     
  5. May 14, 2006 #4

    benorin

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    [tex]\zeta (2m) = (-1)^{n+1}\frac{(2\pi )^{2m}B_{2m}}{2(2m)!}[/tex]

    where [tex]B_n[/tex] is the nth Bernoulli number.
     
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