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Prime gaps asymptotic classification

MatsGranvik Aug 24th, 2018 (edited) 83 Never
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  1. (*start*)
  2. (*Mathematica*)
  3. Clear[nn, h, n, k, m, d, dd];
  4. nn = 20;
  5. h = 2;(*twin primes*)N[
  6.  Table[Sum[
  7.    Sum[If[Mod[n, k] == 0,
  8.       Sum[If[Mod[GCD[n/k, m], d] == 0, MoebiusMu[d]*(d), 0], {d, 1,
  9.          GCD[n/k, m]}]*
  10.        Sum[If[Mod[GCD[k, m + h], c] == 0, MoebiusMu[c]*(c), 0], {c, 1,
  11.           GCD[k, m + h]}], 0], {k, 1, n}]/n, {n, 1,
  12.     nn - h + 100}], {m, 1, nn - h}]];
  13. N[Round[%, 10^-1]]
  14. Table[N[If[n == 1, Log[nn - h + 100] + EulerGamma, MangoldtLambda[n]]*
  15.     MangoldtLambda[n + h]], {n, 1, nn - h}];
  16. N[Round[%, 10^-1]]
  17.  
  18.  
  19. Clear[nn, h, n, k, m];
  20. nn = 17;
  21. h = 2;(*twin primes*)M4 =
  22.  Table[Sum[
  23.    Sum[If[Mod[n, k] == 0,
  24.      Sum[If[Mod[GCD[n/k, m], d] == 0, MoebiusMu[d]*(d), 0], {d, 1,
  25.         GCD[n/k, m]}]*
  26.       Sum[If[Mod[GCD[k, m + h], c] == 0, MoebiusMu[c]*(c), 0], {c, 1,
  27.         GCD[k, m + h]}], 0], {k, 1, n}], {m, 1, n}], {n, 1, nn - h}]
  28.  
  29. Clear[nn, h, n, k, m];
  30. nn = 17 + 2;
  31. h = 4;(*Cousin primes*)M4 =
  32.  Table[Sum[
  33.    Sum[If[Mod[n, k] == 0,
  34.      Sum[If[Mod[GCD[n/k, m], d] == 0, MoebiusMu[d]*(d), 0], {d, 1,
  35.         GCD[n/k, m]}]*
  36.       Sum[If[Mod[GCD[k, m + h], c] == 0, MoebiusMu[c]*(c), 0], {c, 1,
  37.         GCD[k, m + h]}], 0], {k, 1, n}], {m, 1, n}], {n, 1, nn - h}]
  38. (*end*)
  39.  
  40. (*start*)
  41. (*8 Feb 2019*)
  42. TableForm[Table[
  43.   h = 2^hh;
  44.   nn = 82;
  45.   TableForm[CC = Table[
  46.      TableForm[
  47.       A = Table[
  48.         Table[If[Mod[n, k] == 0, MoebiusMu[k]/(k)^(0 - 1), 0], {k, 1,
  49.           n}], {n, 1, nn}]];
  50.      TableForm[
  51.       B = Transpose[
  52.         Table[Table[If[n >= k, A[[n, k]], 0], {k, 1, nn}], {n, 1,
  53.           nn}]]];
  54.      TableForm[
  55.       AA = Table[
  56.         Table[If[Mod[n, k] == 0, B[[n/k, m]], 0], {k, 1, nn}], {n, 1,
  57.          nn}]];
  58.      TableForm[
  59.       BB = Table[
  60.         Table[If[Mod[n, k] == 0, B[[n/k, m + h]], 0], {k, 1, nn}], {n,
  61.           1, nn}]];
  62.      (AA.BB)[[All, 1]], {m, 1, nn - h}]];
  63.   Table[Sum[CC[[n, k]], {n, 1, k}], {k, 1, nn - h}], {hh, 1, 6}]]
  64. (*end*)
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