Friday, November 10, 2017

[vzgekoag] A few notes on the Riemann zeta function

  1. Show that the infinite series Sum(n=1, infinity, 1/n^(1+t*i)) diverges for all t, not just for t=0 which is the harmonic series.  It seems that the curvature induced by the imaginary part might be able to nudge the slow growth to infinity of the harmonic series back down to finiteness; however, this is almost certainly incorrect.  Despite the series presumably diverging, zeta(1+t*i) can be computed by analytic continuation.
  2. Riemann's reflection functional equation for zeta allows calculating zeta(1-s) given the value of zeta(s).  The defining series converges for Re(s)>1, so from that, we can also calculate zeta for Re(s)<0.  The only region we can't calculate is 0<=Re(s)<=1.  But that is the critical strip!  That's where all the interesting stuff happens!  That's where the million dollars lies!  (Obviously there are other formulae to calculate zeta within the critical strip.)
  3. Although the series converges for all Re(s)>1, it converges quite slowly near 1.  This is not too surprising, since it diverges quite slowly at 1.

    ? s=1.01
    ? zeta(s)
    100.58
    ? sum(n=1,1000000,n^(-s))
    time = 7,485 ms.
    13.48

  4. Something I would like to see (this probably exists in a textbook or maybe Riemann's original paper): Here is a expression for zeta which can be used to compute its value for the entire complex plane, and a proofs of analyticity of the expression and that it coincides with the p-series for Re(s)>1.  If the expression is defined piecewise, then we need to show the function remains analytic across the piece boundaries.
  5. The transformation (1-2^(1-s))*zeta(s)=DirichletEta(s) (from Mathworld) (also this YouTube video) does provide an expression which converges for Re(s)>0, thanks to the alternating series.  The rest of the complex plane can be gotten by using the reflection functional equation.  The critical strip can be evaluated both directly through eta and through applying the reflection equation.  This probably explains why zeroes in the critical strip are symmetric around Re(s)=1/2.
  6. Getting zeta(0) using the method above does not straightforwardly work.  Although eta(1)=log 2, one runs into madness like zeta(1) being undefined, log 2=0*zeta(1), and the gamma function being undefined at zero.
  7. The gamma function is also undefined at negative even integers, but I don't think that matters for any other place we would like to use the reflection equation.

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