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%I #94 Mar 05 2020 10:37:29
%S 15,21,25,30,34,37,41,44,47,50,53,56,59,62,64,67,70,72,75,77,80,82,85,
%T 87,90,92,94,97,99,101,103,106,108,110,112,114,117,119,121,123,125,
%U 127,129,131,133,135,137,139,142,144,146,148,150,151,153,155,157,159,161,163
%N Nearest integer to the França-Leclair approximation 2*Pi*(n - 11/8)/LambertW((n - 11/8)/exp(1)) of the Riemann zeta zeros.
%C This sequence is also the nearest integer to the n-th point t on the critical line such that Re(zeta(1/2+i*t))=0 and such that Im(zeta(1/2+i*t)) is not equal to zero, when excluding t=0.819545... Verified for the first 10000 cases. See Mathematica program for how to verify this.
%C Roger Bagula pointed out that the difference between the approximation and the points t, resembles a hyperbola.
%C Compare this sequence to the Gram points A002505.
%C The first point t such that Re(zeta(1/2+i*t))=0 and Im(zeta(1/2+i*t)) is not equal to zero, is: t(1)=14.5179196282622336505419642930... while for n=1 the França-Leclair approximation is 14.5213469530656281679750582094... This gives an error of 0.0034273248033... This decreases to 0.0003990193059... by n=10.
%H G. C. Greubel, <a href="/A273061/b273061.txt">Table of n, a(n) for n = 1..5000</a>
%H Guilherme França and André LeClair, <a href="http://arxiv.org/abs/1407.4358">A theory for the zeros of Riemann Zeta and other L-functions</a>, arXiv:1407.4358 [math.NT], 2014, formula (163) at page 47.
%H Mats Granvik, <a href="/A273061/a273061_1.txt">Mathematica program for the iterative formula.</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/GramPoint.html">Gram Point</a>.
%F a(n) = round(2*Pi*(n - 11/8)/LambertW((n - 11/8)/exp(1))).
%F a(n) = round(2*Pi*exp(1)*exp(LambertW((n - 11/8)/exp(1)))). - _Mats Granvik_, Feb 27 2017
%F a(n) = round(2*Pi*exp(1 + LambertW((8*(n - 3/2) + 1)/(8*e)))) after the formula in MathWorld. - _Mats Granvik_, Feb 25 2017
%F For c = 1/2 the n-th complementary Gram point x is the fixed point solution to the iterative formula: x = 2*Pi*e*e^LambertW(((x/(2*Pi))*log(x/(2*Pi*e)) - c + n - 1 - RiemannSiegelTheta(x)/Pi)/e). - _Mats Granvik_, Jul 24 2017
%t (*The nearest integer to the França-Leclair approximation*)
%t Round[Table[2*Pi*(n - 11/8)/ProductLog[(n - 11/8)/Exp[1]], {n, 1, 60}]]
%t (*The nearest integer to t such that Re(zeta(1/2+I*t))=0 while Im(zeta(1/2+I*t))=/0*)
%t Round[x /. Table[FindRoot[Re[Zeta[1/2 + I*x]] == 0, {x, 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]}], {n, 1, 60}]]
%t Clear[a, n, g]; a[n_] := g /. FindRoot[RiemannSiegelTheta[g] == Pi*(2*n - 1)/2, {g, 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]}]; a = Table[Round[a[n]], {n, 0, 60 - 1}] (* after _Jean-François Alcover_ in A002505 *)
%o (PARI) a(n)=round(2*Pi*exp(lambertw((n-11/8)/exp(1))+1)) \\ Works for n > 1 on GP 2.8.0; _Charles R Greathouse IV_, May 15 2016
%o (Sage)
%o R = RealField(100)
%o a = lambda n: R(2*pi*(n - 11/8)/lambert_w((n - 11/8)/exp(1)))
%o print([a(n).round() for n in (1..60)]) # _Peter Luschny_, May 19 2016
%Y Cf. A002505, A177885, A135297, A273061, A153815, A282793, A282794, A282896, A282897.
%K nonn
%O 1,1
%A _Mats Granvik_, May 14 2016
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