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MatsGranvik

Matrix sum, half- and complete analytic continuation

Nov 20th, 2016
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  1. (*Mathematica 8 start*)
  2. Clear[s, z, c, nn, n, k, kk]
  3. z = N[ZetaZero[1], 20];
  4. c = N[ZetaZero[1], 20];
  5. nn = 64;
  6. A = Table[
  7. Table[If[Mod[n, k] == 0, Sqrt[k]/n^z, 0], {k, 1, nn}], {n, 1,
  8. nn}]; B =
  9. Table[Table[
  10. If[Mod[k, n] == 0, MoebiusMu[n]*Sqrt[n]/k^c, 0], {k, 1, nn}], {n,
  11. 1, nn}];
  12. Print["Matrix sum"]
  13. N[Total[Total[A.B]], 20]
  14. Print["Half analytic continuation"]
  15. N[Sum[Limit[
  16. Zeta[s] Total[1/Divisors[n]^(s - 1)*MoebiusMu[Divisors[n]]],
  17. s -> z]/n^c, {n, 1, nn}], 20]
  18. Print["Complete analytic continuation"]
  19. N[Zeta[z]*Zeta[c]/Zeta[c + z - 1], 20]
  20. Clear[c, ss, t]
  21. ss = 100;
  22. c = 1 + 1/ss;
  23. sigma = 1/2;
  24. Print["Plot of matrix sum on the critical line"]
  25. A = Table[
  26. Table[If[Mod[n, k] == 0, Sqrt[k]/n^(sigma + I*t), 0], {k, 1,
  27. nn}], {n, 1, nn}]; B =
  28. Table[Table[
  29. If[Mod[k, n] == 0, MoebiusMu[n]*Sqrt[n]/k^1, 0], {k, 1, nn}], {n,
  30. 1, nn}];
  31. Plot[Re[Total[Total[A.B]]], {t, 0, 60}]
  32.  
  33. Print["Plot of half analytic continuation on the critical line"]
  34. Plot[Re[Sum[
  35. Zeta[sigma + I*t] Total[
  36. 1/Divisors[n]^(sigma + I*t - 1)*MoebiusMu[Divisors[n]]]/n^1, {n,
  37. 1, 30}]], {t, 0, 60}]
  38.  
  39. Print["Plot of analytic continuation on the critical line"]
  40. Plot[Re[N[Zeta[sigma + I*t]*Zeta[c]/Zeta[c + sigma + I*t - 1],
  41. 20]], {t, 0, 60}, PlotRange -> {0, ss + 2}]
  42. (*Mathematica 8 end*)
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