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MatsGranvik

Riemann zeta zeros multiply truncated geometric series

Jan 7th, 2023 (edited)
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  1. (*start*)
  2. (*Associated Mathematica program*)
  3. Clear[h, x, k, s, s1, nn, m];
  4. $MaxRootDegree = 1000;
  5. m = 10;(*m must be an even integer and greater than 4*)
  6. h = 400;
  7. gcd = GCD @@ Table[Round[Log[n]*h]/h, {n, 1, m}]
  8. Table[Round[Log[n]*h]/h, {n, 1, m}]/gcd
  9. r = 100;
  10. integer = 0;
  11. s = 1/gcd*(2*Pi*I*integer -
  12. Log[Root[
  13. Sum[(-1)^(n + 1) #1^(Round[Log[n]*h]/h/gcd), {n, 1, m}] &, r]])
  14. N[s, 80]
  15. N[Sum[(-1)^(n + 1)/(E^(Round[Log[n]*h]/h))^s, {n, 1, m}]]
  16. s1 = 1/gcd*(2*Pi*I*integer -
  17. Log[Root[
  18. polynomial =
  19. Sum[Sum[#1^k, {k, Round[Log[m - (2*q - 1)]*h]/h/gcd,
  20. Round[Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}] &,
  21. r - 1]])
  22. N[s1, 80]
  23. N[Sum[(-1)^(n + 1)/(E^(Round[Log[n]*h]/h))^s1, {n, 1, m}]]
  24. (*end*)
  25.  
  26. (*start*)
  27. sort = Sort[
  28. Flatten[Table[
  29. Table[k, {k, Round[Log[m - (2*q - 1)]*h]/h/gcd,
  30. Round[Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}]]]
  31. "Plot of exponents of polynomial"
  32. ListLinePlot[sort]
  33. "Plot of coefficients of polynomial"
  34. ListPlot[Sum[
  35. Table[If[sort[[n]] == k, 1, 0], {k, 1, Max[sort]}], {n, 1,
  36. Length[sort]}], Filling -> 0]
  37. (*end*)
  38.  
  39.  
  40. (*start*)
  41. (*m must be an even integer and greater than 4*)
  42. m = 20;
  43. h = 200; sort =
  44. Sort[Flatten[
  45. Table[Table[
  46. k, {k, Round[Log[m - (2*q - 1)]*h]/h/gcd,
  47. Round[Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}]]]
  48. "Plot of exponents of polynomial"
  49. ListLinePlot[sort]
  50. "Plot of coefficients of polynomial"
  51. ListPlot[Sum[
  52. Table[If[sort[[n]] == k, 1, 0], {k, 1, Max[sort]}], {n, 1,
  53. Length[sort]}], Filling -> 0]
  54. (*end*)
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