MatsGranvik

Gram points computed by iterative formula

Jun 18th, 2017
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  1. (* Start *)
  2. (* Gram points to arbitrary precision *)
  3. c = 7/8;
  4. x = N[Table[2*Pi*Exp[1]*Exp[ProductLog[(n - c)/Exp[1]]], {n, 1, 42}],
  5. 40];
  6. Do[n = N[Round[(x/(2*Pi)*Log[x/(2*Pi*Exp[1])] - 1/8 -
  7. RiemannSiegelTheta[x]/Pi) + Range[42], 10^-20], 20];
  8. x = N[Round[2*Pi*Exp[1]*Exp[ProductLog[(n - c)/Exp[1]]], 10^-20],
  9. 20];, {i, 1, 20}]
  10. n
  11. x
  12. RiemannSiegelTheta[x]/Pi
  13. Round[x]
  14.  
  15.  
  16. (* start *)
  17. (* Gram points *)
  18. Table[N[InverseFunction[RiemannSiegelTheta][Pi*(n - 1)], 30], {n, 0,
  19. 12}]
  20. Round[%]
  21. (* end *)
  22.  
  23.  
  24. (* start *)
  25. (* complementary Gram points *)
  26. Table[N[InverseFunction[RiemannSiegelTheta][Pi*(n - 1/2)], 30], {n, 0,
  27. 12}]
  28. Round[%]
  29. (* end *)
  30.  
  31.  
  32. (* Franca-Leclair points to arbitrary precision *)
  33. c = 11/8;
  34. x = N[Table[2*Pi*Exp[1]*Exp[ProductLog[(n - c)/Exp[1]]], {n, 1, 42}],
  35. 40];
  36. Do[n = N[Round[(x/(2*Pi)*Log[x/(2*Pi*Exp[1])] - 1/8 -
  37. RiemannSiegelTheta[x]/Pi) + Range[42], 10^-20], 20];
  38. x = N[Round[2*Pi*Exp[1]*Exp[ProductLog[(n - c)/Exp[1]]], 10^-20],
  39. 20];, {i, 1, 12}]
  40. n
  41. x
  42. RiemannSiegelTheta[x]/Pi
  43. Round[x]
  44. (* End *)
  45.  
  46. (* Start *)
  47. (*Setting the variable start=1 gives the minus 1-th Gram \
  48. point.*)Clear[n]; start = 1; end = 42;(*Interesting values of the \
  49. variable c are:c=0 c=1/2 c=1/4 c=3/4*)c = 0; numberOfIterations = \
  50. 300; decimaldigits = 100; Monitor[
  51. x = Table[x = 1;
  52. Do[x = N[
  53. Round[2*Pi*
  54. E^(ProductLog[-((c - n +
  55. RiemannSiegelTheta[x]/
  56. Pi + (x*(-Log[x] + 1 + Log[2] + Log[Pi]))/(2*
  57. Pi) + 2)/E)] + 1), 10^(-decimaldigits)],
  58. decimaldigits];, {i, 1, numberOfIterations}];
  59. x, {n, start, end}];, n]; Zeta[
  60. 1/2 + I*x]; Round[x] (*_Mats Granvik_,Jun 17 2017*)
  61. (* End *)
  62.  
  63. (* Start *)
  64. (* Mathematica *)
  65. (* Derivation of the Gram point approximation by Weisstein in \
  66. Mathworld, https://oeis.org/A002505 *)
  67. Clear[x, n, a, g];
  68. Series[RiemannSiegelTheta[x], {x, Infinity, 12}]
  69. a = Normal[Series[RiemannSiegelTheta[x], {x, Infinity, 0}]]
  70. g = FullSimplify[(x /. Solve[a == n*Pi, x])[[1]]]
  71. (* g above is equivalent to g[n_] = 2*E^(1 + ProductLog[(1 + \
  72. 8*n)/(8*E)])*Pi *)
  73. n = Range[42] - 2;
  74. N[g, 20]
  75. Round[g]
  76. (* End *)
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