MatsGranvik

Reuns von Mangoldt function convergence Riemann hypothesis

Jun 13th, 2021 (edited)
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  1. (*Mathematica start*)
  2. Monitor[a1 =
  3. Table[-1/n^(1/2)/Log[n]^3 +
  4. Sum[If[a == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}] -
  5. Sum[Sum[If[a*b == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}], {b,
  6. 2, n}] +
  7. Sum[Sum[Sum[
  8. If[a*b*c == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}], {b, 2,
  9. n}], {c, 2, n}] -
  10. Sum[Sum[Sum[
  11. Sum[If[a*b*c*d == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2,
  12. n}], {b, 2, n}], {c, 2, n}], {d, 2, n}], {n, 2, 2^5 - 1}], n]
  13. a2 = Table[(MangoldtLambda[n] - 1)/n^(1/2)/Log[n]^3, {n, 2, 2^5 - 1}]
  14. N[a1 - a2]
  15. Chop[%]
  16. (*end*)
  17.  
  18.  
  19. (*Mathematica start*)
  20. nn = 2^7;
  21. A = Table[
  22. Table[If[Mod[n, k] == 0, ""[n], 0]*If[n > k, 1, 0], {k, 1,
  23. nn}], {n, 1, nn}];
  24. Expand[(MatrixPower[A, 0])[[2^0 ;; 2^1 - 1, 1]]]
  25. Expand[(MatrixPower[A, 1])[[2^1 ;; 2^2 - 1, 1]]]
  26. Expand[(MatrixPower[A, 2])[[2^2 ;; 2^3 - 1, 1]]]
  27. Expand[(MatrixPower[A, 3])[[2^3 ;; 2^4 - 1, 1]]]
  28. Expand[(MatrixPower[A, 4])[[2^4 ;; 2^5 - 1, 1]]]
  29. Expand[(MatrixPower[A, 5])[[2^5 ;; 2^6 - 1, 1]]]
  30. Expand[(MatrixPower[A, 6])[[2^6 ;; 2^7 - 1, 1]]]
  31. (*end*)
  32.  
  33. (*****14.06.2021 klockan 23:54 **********************************************)
  34.  
  35. Limit[1/Sqrt[2^(n - 1)]/Log[2^(n - 1)]^3*
  36. Log[2^(n - 1)]/(n - 1) + (n - 1)/Sqrt[3*2^(n - 1 - 1)]/
  37. Log[3*2^(n - 1 - 1)]^3*Log[3*2^(n - 1 - 1)]/(n - 1), n -> Infinity]
  38.  
  39. 0
  40.  
  41. Limit[1/Sqrt[2^(n - 1)]/Log[2^(n - 1)]^3*
  42. Log[2^(n - 1)]/(n - 1) + (n - 1)/Sqrt[3*2^(n - 1 - 1)]/
  43. Log[3*2^(n - 1 - 1)]^3*
  44. Log[3*2^(n - 1 - 1)]/(n - 1)/(
  45. 1/Sqrt[2^(n + 1 - 1)]/Log[2^(n + 1 - 1)]^3*
  46. Log[2^(n + 1 - 1)]/(n + 1 - 1) + (n + 1 - 1)/
  47. Sqrt[3*2^(n + 1 - 1 - 1)]/Log[3*2^(n + 1 - 1 - 1)]^3*
  48. Log[3*2^(n + 1 - 1 - 1)]/(n + 1 - 1) ), n -> Infinity]
  49.  
  50. Sqrt[2]
  51.  
  52. (***** 14.06.2021 klockan 23:54 **********************************************)
  53.  
  54.  
  55. (* https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown *)
  56.  
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