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A106635
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Rational approximations of Zeta zeros as an integer sequence.
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0
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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ZetaZero[n]=1/2+I*b[n] a(n) = Round[2*b[n]/Pi-4]
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MATHEMATICA
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(* 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
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CROSSREFS
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KEYWORD
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nonn,uned
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AUTHOR
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STATUS
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approved
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