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Zeta function in the critical strip

  1. Dec 15, 2008 #1
    how do i calculate values of the riemann zeta function in the critical strip? because if you only know zeta as a series:

    [tex]
    \zeta(s) = \sum 1/n^s
    [/tex]

    and the functional equation

    [tex]
    \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) \!
    [/tex]

    you can only calculate values that have real part bigger then 1 or smaller then 0.
    i know i can use a math software to calculate it but i want to understand the process.
     
  2. jcsd
  3. Dec 15, 2008 #2
    Hi!,
    there are many other representations (wikipedia or www.mathworld.com) but maybe non of them will be enough helpfull.
     
  4. Dec 19, 2008 #3
    Use the dirichlet eta function relation.
     
  5. Dec 19, 2008 #4
    can we express the eta function as a product of primes?
     
  6. Dec 19, 2008 #5
    in 0< re s <1 ?
     
  7. Dec 20, 2008 #6
  8. Dec 20, 2008 #7
    or, is there a way to calculate values in the critical strip with out using an alternating series?
     
  9. Dec 20, 2008 #8
    Well, you can use the relation to zeta and use its euler product. But I'm not sure as far as the convergence goes.

    edit1: And yes, you can (amongst other ways) express [tex]\eta(s)\Gamma(s)[/tex] as an integral,

    [tex]\eta(s)\Gamma(s)=\int_0^\infty \frac{x^{s-1}}{e^x+1}\mathrm{d}x[/tex], valid for re s > 0.

    and then use the zeta relation again.

    You could also use the [tex]\zeta(s)\Gamma(s)[/tex] integral form, and deform the contour as riemann originally did.
     
    Last edited: Dec 20, 2008
  10. Dec 20, 2008 #9
    i tried using the euler product but it didn't work, but thanks for the eta-gamma integral, can you show me the zeta-gamma integral two and save me the search?
     
  11. Dec 21, 2008 #10
    Just go to almost any gamma or zeta function online encyclopedia site for more info, but beware the original form only works for re s > 1 (the eta form works for re s>0), if you are not somewhat familiar with complex analysis you won't get much of it.

    The eta gamma + relation gives,

    [tex]\zeta(s) = \frac{1}{(1-2^{1-s})\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x+1}\mathrm{d}x[/tex], edit([tex]\Re s > 0, s \not= 1[/tex])
     
  12. Dec 22, 2008 #11
    Thanks!!
     
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