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A100060
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Consider the nontrivial zeros of the Riemann zeta function on the critical line, 1/2 + i*t. a(n) tells where the second difference of the imaginary part is positive (denoted by a 1) or negative (denoted by a 0).
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8
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1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Differences between zeta function gaps: increases are 1 and decreases are 0.
The ratios of the numbers of 0's to the number of 1's in the first 10^n differences are 0/1, 5/5, 50/50, 493/507, 4998/5002, 49949/50049, ...
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REFERENCES
| John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume - a Penguin Group, NY, 2003, pgs. 198-9.
Keith Devlin, The Millennium Problems, The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books, NY, 2002, p. 52.
Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, NY, 2002, pages 93-4, 101.
Marcus du Sautoy, The Music of the Primes, Searching to Solve the Greatest Mystery in Mathematics, Perennial, NY, 2004, pgs. 98-9.
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LINKS
| A. M. Odlyzko, Tables
Index entries for zeta function.
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EXAMPLE
| The first few t values are 14.134725142..., 21.022039639..., 25.010857580...
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MATHEMATICA
| zz = { (* the list of values in the hyper link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; Join[{1}, Table[ If[ yy[[n + 1]] > yy[[n]], 1, 0], {n, 104}]] (* Or *)
zz = { (* the list of values in the hyper link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; xx = Drop[yy, 1] - Drop[yy, -1]; Join[{1}, Table[ If[ xx[[n]] > 0, 1, 0], {n, 104}]] (from Robert G. Wilson v Jan 14 2005)
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CROSSREFS
| Cf. A102522, A102523, A123504, A123505, A123506, A123507.
Sequence in context: A088517 A040053 A004569 * A147850 A099991 A091069
Adjacent sequences: A100057 A100058 A100059 * A100061 A100062 A100063
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 31 2004
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EXTENSIONS
| Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 13 2005
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