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Infinite series convergence

  1. Jun 9, 2009 #1
    [tex]\sum^{\infty}_{x=1} \frac{cos(14.1347 \ln (x))}{x^{a}} = 0[/tex]

    Is there a way to solve for a? I dont think so but maybe someone here will have an insight as to what to do..
  2. jcsd
  3. Jun 25, 2009 #2
    This equation is

    [tex](1/2)\Re \zeta(a-ui) = 0 [/tex]

    for [tex]u = 14.1347[/tex], where [tex]\Re[/tex] signifies the real part, and [tex]\zeta[/tex] is the Riemann zeta function.

    The attached picture shows the graph (I did it without the factor 1/2). The .mw file is the Maple code that generated this picture.

    So [tex]a=1/2[/tex] looks like the solution. If we replace [tex]u=14.1347[/tex] by the nearby zero of the zeta function [tex]u = 14.134725141734693790\cdots[/tex] then the solution would be exactly [tex]a=1/2[/tex] of course.

    Caveat. Probably the original series converges only for [tex]a > 1[/tex] , so my analysis applies only to the analytic continuation.

    Attached Files:

    • zeta.jpg
      File size:
      11.1 KB
    • zeta.mw
      File size:
      29.9 KB
  4. Jun 26, 2009 #3

    OMG brilliant! Thanks! I realized this a few days ago, and just realized now that I realized it, and it looks a bit clearer now. weird.. but Im not complaining, thanks!
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