%I M4924 N2113 #319 Oct 19 2024 18:30:26
%S 14,21,25,30,33,38,41,43,48,50,53,56,59,61,65,67,70,72,76,77,79,83,85,
%T 87,89,92,95,96,99,101,104,105,107,111,112,114,116,119,121,123,124,
%U 128,130,131,133,135,138,140,141,143,146,147,150,151,153,156,158,159,161
%N Nearest integer to imaginary part of n-th zero of Riemann zeta function.
%C "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
%C From _Daniel Forgues_, Jul 24 2009: (Start)
%C 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.
%C 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)
%C The sequence is not strictly increasing. - _Joerg Arndt_, Jan 17 2015
%C 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
%C 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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%H Mats Granvik, <a href="/A002410/a002410.png">Plots from the Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros.</a>
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%H M. R. Watkins, <a href="http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics1.htm">Quantum mechanics: spectral interpretation of Riemann zeros</a>
%H Sebastian Wedeniwski, <a href="http://www.wstein.org/simuw/misc/zeta_grid.html">ZetaGrid</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannHypothesis.html">Riemann Hypothesis</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Xi-Function.html">Xi-Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_hypothesis">Riemann hypothesis</a>
%H Wolfram Research, <a href="http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/30/01">First few computations of Z(n) (nontrivial zeros power sums)</a>
%H A. Zaccagnini, <a href="http://people.dmi.unipr.it/alessandro.zaccagnini/psfiles/papers/Q429.pdf">Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function</a>
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>
%F 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
%F 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
%e The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).
%t Table[Round[Im[ZetaZero[n]]], {n, 59}] (* _Jean-François Alcover_, May 02 2011 *)
%o (Sage)
%o def A002410_list(n):
%o Z = lcalc.zeros(n)
%o return [round(z) for z in Z]
%o A002410_list(59) # _Peter Luschny_, May 02 2014
%o (PARI) apply(round,lfunzeros(lzeta,100)) \\ _Charles R Greathouse IV_, Mar 10 2016
%Y Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).
%Y 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).
%K nonn,easy,nice,changed
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004