A106635 Rational approximations of Zeta zeros as an integer sequence.

5, 9, 12, 15, 17, 20, 22, 24, 27, 28, 30, 32, 34, 35, 37, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 52, 55, 56, 57, 59

Offset

0,1

Comments

The average error in the approximation is low (-0.0559729 )for the first 30 zeta zeros. The idea is that the imaginary part of the zeta zero is a bad rational approximation of the type: 4/(a(n)+4) to give b[n]=2*Pi*(a(n)+4)/4

Links

Table of n, a(n) for n=0..29.

Formula

ZetaZero[n]=1/2+I*b[n] a(n) = Round[2*b[n]/Pi-4]

Mathematica

(* Zeta Zero data from FindRoot *) z[0] = 0.5+ 14.1347251293669318` I; z[1] = 0.5+ 21.0220394302515201` I; z[2] = 0.5+ 25.0108575810860855` I; z[3] = 0.5+ 30.4248761260581268` I; z[4] = 0.5+ 32.935061587739185`I; z[5] = 0.5 + 37.5861781581133191` I; z[6] = 0.5+ 40.9187190119943267` I; z[7] = 0.5+ 43.327073280866557` I; z[8] = 0.5+ 48.0051508794884007` I; z[9] = 0.5 + 49.7738324775595852` I; z[10] = 0.5+ 52.97032147771447` I; z[11] = 0.5+ 56.44624769706339` I; z[12] = 0.5+ 59.34704400260235` I; z[13] = 0.5 + 60.8317785246098` I; z[14] = 0.5+ 65.1125440480816` I; z[15] = 0.5+ 65.1125440480816` I; z[16] = 0.5+ 67.07981052949417` I; z[17] = 0.5 69.54640171117398` I; z[18] = 0.5+ 72.0671576744819` I; z[19] = 0.5+ 75.70469069908393` I; z[20] = 0.5 + 77.14484006887483` I; z[21] = 0.5+ 79.33737502024937` I; z[22] = 0.5 + 82.91038085408603` I; z[23] = 0.5+ 84.73549298051705` I; z[24] = 0.5+ 87.42527461312524` I; z[25] = 0.5+ 87.42527461312524` I; z[26] = 0.5 + 92.49189927055848` I; z[27] = 0.5 + 94.65134404051992` I; z[28] = 0.5 + 95.87063422824534` I; z[29] = 0.5+ 98.8311942181937` I; a = Table[Round[2*Im[z[n]]/Pi - 4], {n, 0, 29}] averageError=Apply[Plus, Table[2*Im[z[n]]/Pi - 4 - Round[2*Im[z[n]]/Pi - 4], {n, 0, 29}]]/30

Crossrefs

Sequence in context: A356462 A102183 A153044 * A314614 A182617 A314615

Adjacent sequences: A106632 A106633 A106634 * A106636 A106637 A106638

Keyword

nonn,uned

Author

Roger L. Bagula, May 11 2005

Status

approved