1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Riemann Zeta Function shows non-trival zeros critical-strip symmetry

  1. Jan 10, 2017 #1
    1. The problem statement, all variables and given/known data

    I want to show that the non-trival zeros of the Riemann Zeta function all lie in the critical strip ## 0 < Re(s) < 1## and further to this that they are symmetric about the line ##Re(s)= 1/2 ##

    where ## \zeta(s) = \sum\limits^{\infty}_{n=1}n^{-s}##

    With the functional equation ##\zeta(1-s)=\pi^{1/2 - s} \frac{\Gamma(s/2)}{\Gamma(1/2(1-s))} \zeta (s) ##

    2. Relevant equations

    see above

    3. The attempt at a solution

    My first step is to narrow down to ##Re(s) < 1 ## since for ##Re(s)>0## the following holds:

    ## \zeta (s) = \Pi_{p} \frac{1}{1-p^{-s}} \neq 0 ## since no terms are zero.

    Then using the functional equation ##\zeta(1-s)=\pi^{1/2 - s} \frac{\Gamma(s/2)}{\Gamma(1/2(1-s))} \zeta (s) ## and sending ##s=-2k## and making ##\zeta(-2k)## the subject, it is clear that the non-trival zeros occur at ##s=-2k## arising from the reciprocal of the poles of ##\Gamma(s)## for ##s=-2,-4...##

    I also have that ##\zeta(s)## has a pole at ##s=1## and ##\zeta(0)=-1/2## so I now have the critical strip as ##0< Re(s) < 1## .

    I am now stuck what to do how to prove that they are symmetric about ##s=1/2##. Any hints on getting started, help, greatly appreciated.

    Many thanks
     
  2. jcsd
  3. Jan 10, 2017 #2

    fresh_42

    Staff: Mentor

    Without being in the subject, Wikipedia provides some hint concerning the symmetry at ##Re(s)=\frac{1}{2}.##
    I haven't checked any details, but it might help here. The link to the original "paper" is:
    https://de.wikisource.org/wiki/Über_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Größe
    (Wrong language, of course, but maybe you can read it anyway or think along the formulas.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Riemann Zeta Function shows non-trival zeros critical-strip symmetry
Loading...