

A002410


Nearest integer to imaginary part of nth zero of Riemann zeta function.
(Formerly M4924 N2113)


45



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

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


REFERENCES

Gregory Benford, Gravity's whispers, Futures Column, Nature, 446 (15 July 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. 107128 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.
A. Y. Cheer & D. A. Goldston, "Simple Zeros of the Riemann ZetaFunction" in 'Proceed. of The Amer. Math. Soc.' pp. 365372 vol. 118 No. 2, 1993.
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. 1962) 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.
R. Garunkstis and J. Steuding, Questions around the Nontrivial Zeros of the Riemann ZetaFunction. Computations and Classifications, Math. Model. Anal. 16 (2011), 7281.
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 ZetaFunction: 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 ZetaFunction, Walter de Gruyter, Berlin 1992.
G. Lachaud, "L'hypothese de Riemann" in La Recherche No.346 October 2001 pp. 2430 (or Les Dossiers de La Recherche No. 20 August 2005 pp. 2635) Paris.
M. L. Lapidus, In Search of the Riemann Zeros, Amer. Math. Soc. (AMS) Providence RI 2008.
P. Meier & J. Steuding, "L'hypothese de Riemann" in 'Pour la Science' (French Edition of 'Scientific American') pp 229, March 2009, Issue No. 377, Paris. [From Lekraj Beedassy, Apr 08 2009]
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 ZetaFunction, 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).
K. Soundararajan, On the Distribution of Gaps between Zeros of the Riemann Zeta Function, in 'The Quarterly Journal of Mathematics' pp. 3837 Sep. 1996 Vol. 47 No. 187 Oxford Univ. Press.
E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, Clarendon Press NY 1986.
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.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. AriasdeReyna, XRay of Riemann's zetafunction
J. AriasdeReyna, XRay Of Riemann's ZetaFunction(Part 2)
E. Bogomolny et al., On the spacing distribution of the Riemann zeros:corrections to the asymptotic result
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
P. Borwein et al., The Riemann Hypothesis
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 ZetaFunction 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
T. H. Chan, Pair correlation of the zeros of the Riemann zeta function in longer ranges
H. T. Chan, Distribution of the zeros of the Riemann zeta function in longer intervals
H. T. Chan, More precise Pair Correlation Conjecture
H. T. Chan, More precise pair correlation of zeros and primes in short intervals
Chance News, Chance in the Primes
A. Y. Cheer & D. A. Goldston, Simple Zeros of the Riemann ZetaFunction, Proc. Am. Math. Soc. 118 (1993) 365
Y.J. Choie et al., On Robin's criterion for the Riemann Hypothesis
B. Cipra, A Prime Case of Chaos
J. B. Conrey, The Riemann Hypothesis
J. B. Conrey & G. Myerson, On the BalazardSaias criterion for the Riemann Hypothesis
J. Noel Cook, On Neutronic Functions and Undefined Figures in Prime Distribution [From Lekraj Beedassy, Jan 12 2009]
E. S. Croot, Notes on the Riemann Zeta Function (Functional Equation)
C. Daney, Open Questions:The Riemann Hypothesis
H. Delille, L'Hypothese de Riemann(Expository papers in French) [popup windows]
J. Derbyshire, Prime Obsession [?Broken link]
E. Elizalde, V. Moretti & S. Zerbini, On recent strategies proposed for proving Riemann hypothesis
D. W. Farmer, Counting distinct zeros of the Riemann zetafunction
K. Ford & A Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function
W. F. Galway, Computations related to the Riemann Hypothesis
R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zetafunction, Math. Comp. 76 (2007), 323337.
D. A. Goldston, Notes on Pair Correlation of Zeros and Prime Numbers
D. A. Goldston & S. M. Gonek, A note on S(T) and the zeros of the Riemann zetafunction
S. Gonek, Three Lectures on the Riemann ZetaFunction
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 [From Lekraj Beedassy, Nov 21 2008]
X. Gourdon & P. Sebah, The Riemann Zetafunction 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 zeros de la fonction zeta(s) de Riemann, Acta Mathematica, 27 (1903), 289304.
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
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. HeathBrown, Zeros of the Riemann ZetaFunction on the Critical Line
A. Iqbal, Prime Numbers and Riemann Zeta Function
A. Ivic, On some reasons for doubting the Riemann hypothesis
A. Ivic, On some recent results in the theory of the zetafunction
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
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 Nth zero at large N, arXiv:1305.2613v4 [mathph]
N. Levinson, At Least OneThird of Zeros of Riemann's ZetaFunction are on a=1/2
P. Li, On Montgomery's Pair Correlation Conjecture to the Zeros of the Riemann Zeta Function
Lionman & Allispaul, Riemann Hypothesis
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 firstdigit frequencies of prime numbers and Riemann zeta zeros, Proceedings of The Royal Society A, April 22 2009. [From Lekraj Beedassy, May 10 2009]
K. Maslanka, Nontrivial zeros of the zetafunction of Riemann
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
N. Ng, Large Gaps Between The Zeros Of The Riemann Zeta Function
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
Oklahoma State Mathematics Department, The Riemann zetafunction
Ed. Pegg Jr., Ten Trillion Zeta Zeros
Ed. Pegg Jr., The Riemann Hypothesis
A. Peretti, The Riemann Hypothesis
J. C. PerezMoure, Evidences that the Riemann Hypothesis is true
J. Perry, Riemann's Hypothesis
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 13/28)
H. J. J. te Riele, Some historical and other notes about the Mertens conjecture and its recent disproof
H. J. J. te Riele, Computing the Riemann hypothesis(Text in Dutch)
H. J. J. te Riele, On the History of the function M(x)/sqrt(x) since Stieltjes
A. Rifat, A Physicsbased 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
A. M. Selvam, Signatures of quantumlike chaos in spacing intervals of nontrivial Riemann zeta zeros and in turbulent fluid flows
A. M. Selvam, Signatures of Quantumlike Chaos in Spacing Intervals of Nontrivial Riemann Zeta Zeros and in Turbulent Fluid Flows
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), 421424.
J. Sondow, The Riemann Hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums
J. Sondow, MathWorld: Riemannvon 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), 3740.
G. SpencerBrown, A Short Proof Of Riemann's Hypothesis
Jörn Steuding, On simple zeros of the Riemann zetafunction
E. C. Titchmarsh, The Zeroes of the Riemann ZetaFunction, Proc. Royal Soc. London, 151 pp. 234255 1935.
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), 667681.
B. Van der Pol, An ElectroMechanical Investigation Of The Riemann ZetaFunction In The Critical Strip
M. Verma, Zeros of the Riemann ZetaFunction
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, 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, The first 50 billion zeros of the Riemann zeta function
Eric Weisstein's World of Mathematics, Riemann Hypothesis
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Eric Weisstein's World of Mathematics, XiFunction
Wikipedia, Riemann hypothesis
Wolfram Research, First few computations of Z(n) (nontrival zeros power sums)
A. Zaccagnini, Primes in almost all short intervals and the distribution of the zeros of the Riemann zetafunction
Index entries for zeta function.


FORMULA

a(n) ~ 2 Pi n/log n.  Charles R Greathouse IV, Sep 14 2012
a(n) ~ 2*Pi*exp(1)*(n  11/8)/exp(1)/ProductLog((n  11/8)/exp(1)). This is André LeClair's approximation.  Mats Granvik, Mar 10 2015


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}] (* JeanFrançois 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


CROSSREFS

Cf. A013629, A058303, A057641, A057640, A058209, A058210, A092783, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885.
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



