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A106636 Pair rational approximations of Zeta zeros as an integer sequence. 0
9, 13, 16, 19, 21, 24, 26, 28, 31, 32, 34, 36, 38, 39, 41, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 56, 59, 60, 61, 63 (list; graph; refs; listen; history; text; internal format)
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 Pair rational is ( this sequence) : a(n)/(a(n)+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]

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 ], {n, 0, 29}] averageError=Apply[Plus, Table[2*Im[z[n]]/Pi - Round[2*Im[z[n]]/Pi ], {n, 0, 29}]]/30

CROSSREFS

Sequence in context: A065977 A294999 A153051 * A186427 A050109 A294357

Adjacent sequences:  A106633 A106634 A106635 * A106637 A106638 A106639

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, May 11 2005

STATUS

approved

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Last modified August 14 13:51 EDT 2020. Contains 336481 sequences. (Running on oeis4.)