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Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function. (Formerly M4924 N2113) 59 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 (list; graph; refs; listen; history; text; internal format) OFFSET 1,1 COMMENTS "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 From Daniel Forgues, Jul 24 2009: (Start) 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 REFERENCES 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. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 J. Arias-de-Reyna, X-Ray of Riemann's zeta-function, arXiv:math/0309433 [math.NT], 2003. E. Bogomolny et al., On the spacing distribution of the Riemann zeros:corrections to the asymptotic result, arXiv:math/0602270 [math.NT], 2006. E. Bogomolny et al., On the spacing distribution of the Riemann zeros: corrections to the asymptotic result, Journal of Physics A: Mathematical and General, Vol. 39, No. 34 (2006), 10743-10754. E. Bombieri, The Riemann Hypothesis E. Bombieri, The Indivisible Man: Reviews of "Prime Obsession" by J. Derbyshire & "The Riemann Hypothesis" by K. Sabbagh J. M. Borwein, D. M. Bradley & R. E. Crandall, Computational strategies for the Riemann zeta function L. de Branges, Apology For The Proof of The Riemann Hypothesis R. P. Brent, J. van de Lune, H. J. J. te Riele & D. T. Winter, The first 200,000,001 zeros of Riemann's zeta function K. A. Broughan, Encoding of and Phase portraits of the Riemann Zeta Zeros C. K. Caldwell, The Prime Glossary, Riemann hypothesis C. S. Calude et al., Do the Zeros of Riemann's Zeta-Function Form a Random Sequence ? J. Case, Bringing the Riemann Zeta Function to the World's Attention: Review of "The Music of the Primes" by M. du Sautoy T. H. Chan, Pair Correlation of the zeros of the Riemann zeta function in longer ranges, arXiv:math/0305340 [math.NT], 2003. H. T. Chan, Distribution of the zeros of the Riemann zeta function in longer intervals, arXiv:math/0305341 [math.NT], 2003. H. T. Chan, More precise Pair Correlation Conjecture, arXiv:math/0206293 [math.NT], 2002. H. T. Chan, More precise pair correlation of zeros and primes in short intervals, arXiv:math/0206292 [math.NT], 2002. Chance News, Chance in the Primes A. Y. Cheer & D. A. Goldston, Simple Zeros of the Riemann Zeta-Function, Proc. Am. Math. Soc. 118 (1993) 365. Y.-J. Choie et al., On Robin's criterion for the Riemann Hypothesis, arXiv:math/0604314 [math.NT], 2006. Y.-J. Choie et al., On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol 19, No. 2 (2007), 357-372. B. Cipra, A Prime Case of Chaos J. B. Conrey, The Riemann Hypothesis, Notices of the AMS, Vol. 50, No. 3 (2003), 341-353. J. B. Conrey & G. Myerson, On the Balazard-Saias criterion for the Riemann Hypothesis, arXiv:math/0002254 [math.NT], 2000. E. S. Croot, Notes on the Riemann Zeta Function (Functional Equation) C. Daney, Open Questions:The Riemann Hypothesis H. Delille, L'Hypothèse de Riemann (Expository papers in French) [popup windows] J. Derbyshire, Prime Obsession E. Elizalde, V. Moretti & S. Zerbini, On recent strategies proposed for proving Riemann hypothesis, arXiv:math-ph/0109006, 2001. E. Elizalde, V. Moretti, and S. Zerbini, On Recent Strategies Proposed for Proving the Riemann Hypothesis, International Journal of Modern Physics A, Vol. 18, No. 12 (2003), 2189-2196. D. W. Farmer, Counting distinct zeros of the Riemann zeta-function, The Electronic Journal of Combinatorics, 2 (1995), #R1. K. Ford & A Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, arXiv:math/0405459 [math.NT], 2004. K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, Journal für die reine und angewandte Mathematik, Vol. 2005, No. 579 (2005), 145-158. W. F. Galway, Computations related to the Riemann Hypothesis R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zeta-function, Math. Comput. 76 (2007), 323-337. R. Garunkstis and J. Steuding, Questions around the Nontrivial Zeros of the Riemann Zeta-Function. Computations and Classifications, Math. Model. Anal. 16 (2011), 72-81. D. A. Goldston, Notes on Pair Correlation of Zeros and Prime Numbers, arXiv:math/0412313 [math.NT], 2004. D. A. Goldston & S. M. Gonek, A note on S(T) and the zeros of the Riemann zeta-function, arXiv:math/0511092 [math.NT], 2005. S. Gonek, Three Lectures on the Riemann Zeta-Function, arXiv:math/0401126 [math.NT], 2004. J. Good & B. Churchhouse, A New Conjecture Related to the Riemann Hypothesis X. Gourdon, The 10^13 first zeros of the Riemann Zeta function and zeros computation at very large height X. Gourdon & P. Sebah, The Riemann Zeta-function zeta(s) S. W. Graham, Review of "Prime Obsession" by J. Derbyshire S. W. Graham, Review of "The Riemann Hypothesis" by K. Sabbagh J. P. Gram, Note sur les zéros de la fonction zeta(s) de Riemann, Acta Mathematica, 27 (1903), 289-304. Mats Granvik, How plot the Riemann zeta zero spectrum with the Fourier transform in Mathematica? Mats Granvik, Illustration of Andre LeClair's approximation of the initial terms Mats Granvik, Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros. Mats Granvik, Plots from the Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros. A. Granville, Nombres premiers et chaos quantique (Text in French) J. Hadamard, Sur la distribution des zeros de la fonction zeta(s) et ses consequences arithmetiques (Text in French) B. Hayes, The Spectrum of Riemannium D. R. Heath-Brown, Zeros of the Riemann Zeta-Function on the Critical Line A. Ivic, On some reasons for doubting the Riemann hypothesis, arXiv:math/0311162 [math.NT], 2003. A. Ivic, On some recent results in the theory of the zeta-function, arXiv:math/0312425 [math.NT], 2003. A. Ivic & H. J. J. te Riele, On the zeros of the error term for the mean square of | zeta(1/2 + it) | D. Jao, PlanetMath.Org, Riemann zeta function C.-X. Jiang, Disproofs Of Riemann's Hypothesis N. M. Katz & P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), 1-26. E. Klarreich, Prime Time A. F. Lavrik, Riemann hypotheses A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613 [math-ph], 2013. N. Levinson, At Least One-Third of Zeros of Riemann's Zeta-Function are on sigma=1/2, Proc Natl Acad Sci U S A. 1974 Apr; 71(4): 1013-1015. P. Li, On Montgomery's Pair Correlation Conjecture to the Zeros of the Riemann Zeta Function Lionman & Allispaul, Riemann Hypothesis LMDBF, The Riemann zeta-function J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation Of Zeros Of Riemann's Zeta Function J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation Of Zeros Of Riemann's Zeta Function, II J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation Of Zeros of Riemann's Zeta Function, III J. van de Lune & H. J. J. te Riele, On the Zeros of the Riemann Zeta Function in the Critical Strip, III J. van de Lune, H. J. J. te Riele & D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip, IV B. Luque & L. Lucasa, The first-digit frequencies of prime numbers and Riemann zeta zeros, Proceedings of The Royal Society A, Apr 22 2009. J. H. Mathews, The Riemann Hypothesis (Bibliography) R. Meyer, A Spectral Interpretation for The Zeros of The Riemann Zeta Function N. Ng, Large gaps between the zeros of the Riemann zeta function, arXiv:math/0510530 [math.NT], 2005. A. M. Odlyzko, Tables of zeros of the Riemann zeta function A. M. Odlyzko, Primes, Quantum Chaos and Computers A. M. Odlyzko, On The Distribution Of Spacings Between Zeros Of The Zeta Function A. M. Odlyzko, Papers on Zeros of the Riemann Zeta Function and Related Topics A. M. Odlyzko & H. J. J. te Riele, Disproof of the Mertens Conjecture A. M. Odlyzko & M. Schoenhage, Fast algorithms for multiple evaluations of the Riemann zeta function Ed. Pegg Jr., Ten Trillion Zeta Zeros Ed. Pegg Jr., The Riemann Hypothesis J. C. Perez-Moure, Evidences that the Riemann Hypothesis is true Simon Plouffe, The first (non trivial) zeros of the Riemann Zeta function G. Pugh, The Riemann Hypothesis in a Nutshell K. Ramachandra, 'Current Science' 77(7)951 10.10.1999, On the future of Riemann Hypothesis(pp 1-3/28) H. J. J. te Riele, Some historical and other notes about the Mertens conjecture and its recent disproof H. J. J. te Riele, On the History of the function M(x)/sqrt(x) since Stieltjes A. Rifat, A Physics-based Explanation of the Riemann Hypothesis and its Relationship to Signal Processing P. Sarnak, Problems of the Millennium : The Riemann Hypothesis (2004) P. Sarnak, Review of "The Riemann Zeta Function" by A. A. Karatsuba & S. M. Voronin M. du Sautoy, The music of the primes N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 6. J. Sondow, Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series, Proc. Amer. Math. Soc. 120 (1994), 421-424. J. Sondow, The Riemann Hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums J. Sondow, MathWorld: Riemann-von Mangoldt Formula J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Period. Math. Hungar. 60 (2010), 37-40. K. Soundararajan, On the Distribution of Gaps between Zeros of the Riemann Zeta Function, in 'The Quarterly Journal of Mathematics' pp. 383-Sep 07 1996 Vol. 47 No. 187 Oxford Univ. Press. G. Spencer-Brown, A Short Proof Of Riemann's Hypothesis Jörn Steuding, On simple zeros of the Riemann zeta-function E. C. Titchmarsh, The Zeroes of the Riemann Zeta-Function, Proc. Royal Soc. London, 151 pp. 234-255 1935. J. van de Lune, H. J. J. te Riele and D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann's Zeta Function, Report NW 113/81, Mathematical Centre, Amsterdam, October 1981. J. van de Lune, H. J. J. te Riele and D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip IV, Mathematics of Computation 46 (1986), 667-681. B. Van der Pol, An Electro-Mechanical Investigation Of The Riemann Zeta-Function In The Critical Strip, Bull. Amer. Math. Soc. 53 (1947), 976-981. A. de Vries, The Graph of the Riemann zeta function zeta(s) A. de Vries, The Mystery of the Land of Riemannia M. R. Watkins, proposed (dis)proofs of the Riemann Hypothesis M. R. Watkins, The Riemann Hypothesis M. R. Watkins, A selection of quotations on primes distribution, Riemann zeta function and Riemann hypothesis M. R. Watkins, The 'encoding' of the distribution of prime numbers by the nontrivial zeros of the Riemann zeta function M. R. Watkins, Quantum mechanics: spectral interpretation of Riemann zeros Sebastian Wedeniwski, ZetaGrid Eric Weisstein's World of Mathematics, Riemann Hypothesis Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros Eric Weisstein's World of Mathematics, Xi-Function Wikipedia, Riemann hypothesis Wolfram Research, First few computations of Z(n) (nontrivial zeros power sums) A. Zaccagnini, Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function Index entries for zeta function FORMULA a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Charles R Greathouse IV, Sep 14 2012, corrected by Hal M. Switkay, Oct 04 2021 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 EXAMPLE The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). MATHEMATICA Table[Round[Im[ZetaZero[n]]], {n, 59}] (* Jean-François Alcover, May 02 2011 *) PROG (Sage) def A002410_list(n): Z = lcalc.zeros(n) return [round(z) for z in Z] A002410_list(59) # Peter Luschny, May 02 2014 (PARI) apply(round, lfunzeros(lzeta, 100)) \\ Charles R Greathouse IV, Mar 10 2016 CROSSREFS Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation). Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10). Sequence in context: A013629 A234802 A162780 * A108606 A129497 A255742 Adjacent sequences: A002407 A002408 A002409 * A002411 A002412 A002413 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004 STATUS approved

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Last modified August 27 16:23 EDT 2024. Contains 375470 sequences. (Running on oeis4.)