# 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.