Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    \zeta(s) = \sum 1/n^s

    and the functional equation

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

    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
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook