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MatsGranvik

limiting ratio Golden Ratio minus 1 same program for zeta zeros

Jul 27th, 2020
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  1. (*Program 2*)
  2. (*The limiting ratio of the form (n-1)*a (n-1)/a(n) applied to the \
  3. first column of the matrix inverse of the binomial matrix multiplied \
  4. elementwise with the coefficients in the Taylor polynomial of the \
  5. Riemann zeta function,expanded at an arbitrary point in the critical \
  6. strip (with imaginary part greater than 10),appended with trailing \
  7. zeros.The trailing zeros...0,0,0,0,0,0,0,... in the vector "a" give \
  8. better convergence to the Riemann zeta zeros*)
  9. (*start*)
  10. Clear[start, end, realPart, zz, d, digits, a, nn, A, n, k, b, z, list,
  11. x];
  12. start = 10;
  13. end = 60;
  14. realPart = 0;
  15. (*realPart=1/2;*)
  16. (*realPart=1;*)
  17. (*d must be much greater than zz*)
  18. Monitor[list = Table[zz = 20;
  19. d = 40;
  20. digits = 30;
  21. a = Flatten[{CoefficientList[
  22. Normal[Series[(1 - x - x^2), {x, 0, zz}]], x]*Range[0, 2]!,
  23. Range[d]*0}];
  24. nn = Length[a];
  25. A = Table[
  26. Table[If[n >= k, Binomial[n - 1, k - 1]*a[[n - k + 1]], 0], {k,
  27. 1, Length[a]}], {n, 1, Length[a]}];
  28. Quiet[b = Inverse[A][[All, 1]]];
  29. 0*z*I + N[(nn - 1)*b[[nn - 1]]/b[[nn]], digits], {z, start, end,
  30. 1/10}], z*10]
  31. "Plot of the real part:"
  32. ListLinePlot[Re[list], PlotRange -> {-1, 3}, DataRange -> {start, end}]
  33. "Plot of the imaginary part:"
  34. ListLinePlot[Im[list], DataRange -> {start, end}]
  35. (*end*)
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