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A002410
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Nearest integer to imaginary part of n-th zero of Riemann zeta function.
(Formerly M4924 N2113)
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59
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14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, 140, 141, 143, 146, 147, 150, 151, 153, 156, 158, 159, 161
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OFFSET
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1,1
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COMMENTS
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"All these zeros of the form s + it have real part s = 1/2 and are simple. Thus the Riemann hypothesis is true at least for t < 3330657430697." - Wedeniwski
All nontrivial zeros on the critical line, of the form 1/2 + i*t, have an associated conjugate nontrivial zero of the form 1/2 - i*t.
Any nontrivial zeros off the critical line, if ever found, would come in pairs (1/2 +- delta) + i*t, 0 < delta < 1/2. Each of these pairs, again if ever found, would then have their associated conjugate pair (1/2 +- delta) - i*t, 0 < delta < 1/2. (End)
The sequence is not strictly increasing. - Joerg Arndt, Jan 17 2015
The fraction of numbers n such that a(n) = a(n-1) has density 1. There are only finitely many numbers n with a(n) > a(n-1) + 1, see A208436. - Charles R Greathouse IV, Mar 07 2018
Conjecture: Noninteger rationals of the form m/2^bigomega(m) that can be used to approximate this sequence, i.e. a(n) ~~ 2*Pi*A374074(n)/2^bigomega(A374074(n)) - n/2 +- (...), where '~~' means 'close to'. - Friedjof Tellkamp, Jul 04 2024
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REFERENCES
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Gregory Benford, Gravity's whispers, Futures Column, Nature, 446 (Jul 15 2010), p. 406. [Gravity waves are detected on Earth that turn out to contain a list of the zeros of the Riemann zeta function, essentially this sequence]
E. Bombieri, "The Riemann Hypothesis" in 'The Millennium Prize Problems' Chap. 7 pp. 107-128 Eds: J. Carlson, A. Jaffe & A. Wiles, Amer. Math. Soc. Providence RI 2006.
P. Borwein et al., The Riemann Hypothesis, Can. Math. Soc. (CMS) Ottawa ON 2007.
S. Chowla, Riemann Hypothesis and Hilbert's Tenth Problem, Mathematics and Its Application Series Vol. 4, Taylor & Francis NY 1965.
J. Derbyshire, Prime Obsession, Penguin Books 2004.
K. Devlin, The Millennium Problems, Chapter 1 (pp. 19-62) Basic Books NY 2002.
M. du Sautoy, The Music of the Primes, Fourth Estate/HarperCollins NY 2003.
H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.
A. Ivic, The Riemann Zeta-Function: Theory and Applications, Dover NY 2003.
D. S. Jandu, Riemann Hypothesis and Prime Number Theorem, Infinite Bandwidth Publishing, N. Hollywood CA 2006.
A. A. Karatsuba & S. M. Voronin, The Riemann Zeta-Function, Walter de Gruyter, Berlin 1992.
G. Lachaud, "L'hypothèse de Riemann" in La Recherche No.346 October 2001 pp. 24-30 (or Les Dossiers de La Recherche No. Aug 20 2005 pp. 26-35) Paris.
M. L. Lapidus, In Search of the Riemann Zeros, Amer. Math. Soc. (AMS) Providence RI 2008.
P. Meier & J. Steuding, "L'hypothèse de Riemann" in 'Pour la Science' (French Edition of 'Scientific American') pp 22-9, March 2009, Issue No. 377, Paris.
P. Odifreddi, The Mathematical Century, Chapter 5.2, p. 168, Princeton Univ. Press NJ 2004.
S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Univ. Press, UK 1995.
D. N. Rockmore, Stalking the Riemann Hypothesis, Jonathan Cape UK 2005.
K. Sabbagh, The Riemann Hypothesis, Farrar Straus Giroux NY 2003.
K. Sabbagh, Dr. Riemann's Zeros, Atlantic Books London 2003.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press NY 1986.
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LINKS
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FORMULA
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a(n) ~ 2*Pi*(n - 11/8)/ProductLog((n - 11/8)/exp(1)). This is the asymptotic by Guilherme França and André LeClair. - Mats Granvik, Mar 10 2015; corrected May 16 2016
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EXAMPLE
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The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).
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MATHEMATICA
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PROG
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(Sage)
Z = lcalc.zeros(n)
return [round(z) for z in Z]
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CROSSREFS
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Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004
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STATUS
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approved
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