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Reimann veta function ~1

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  • Shane
    Evaluation of the sum of reciprocals, and the product of linear q. [1] ~ [1/(1^s)+1/(2^s)+1/(3^s)+1/(5^s)+1/(8^s)+1/(13^s)+...] ~ [(2^s)/(2^s)-1 *
    Message 1 of 1 , Oct 23, 2002
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      Evaluation of the sum of reciprocals, and the product of linear q.


      [1] ~

      [1/(1^s)+1/(2^s)+1/(3^s)+1/(5^s)+1/(8^s)+1/(13^s)+...] ~

      [(2^s)/(2^s)-1 * (5^s)/(5^s)-1 * (13^s)/(13^s)-1 *...]



      When s is small, these are farthest apart.(I believe 1/1 with s=1)
      (Thanks to David Broadhurst)
      When s is large, they converge.
      Here are approx. rates of convergence
      s=1 1/1???????
      s=2 1/36.63
      s=3 1/50.103
      s=4 1/120.25
      s=5 1/319.75
      s=6 1/887.68
      s=7 1/2524.56
      s=8 1/7291.44
      s=9 1/21281.12
      s=10 1/62574.30
      s=11 1/184945.44
      s=12 1/548847.42
      s= infinity......

      (do you see a ratio emerging, with succesive s ?)


      BTW, here is a list of q again, if you want to try it yourself:
      q=2,5,13,37,73,89,113,149,157,193,233,269,277,313,353,389,397,457,557,
      613,673,677,733,757,877,953,977,997,1069.....
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