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 A288640 a(n) = (1 + sign(Im(ZetaZero(n)) - 2*Pi*e*exp(LambertW((n - 11/8)/e))))/2. 3
 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS 2*Pi*e*exp(LambertW((n - 11/8)/e)) is the Franca-Leclair asymptotic of the nontrivial Riemann zeta zeros. Positions of 0 are found in A282897. Positions of 1 are found in A282896. LINKS G. C. Greubel, Table of n, a(n) for n = 1..5000 Guilherme França and André LeClair, A theory for the zeros of Riemann Zeta and other L-functions, arXiv:1407.4358 [math.NT], 2014, formula (163) at page 47. Mats Granvik, Mathematica programs to compute the sequence. FORMULA Let ZetaZero(k) denote the zero of the Riemann zeta function on the critical line which has the k-th smallest positive imaginary part. a(n) = (1 + sign(Im(ZetaZero(n)) - 2*Pi*e*exp(LambertW((n - 11/8)/e))))/2. a(n) ~ (floor(Im(ZetaZero(n))/(2*Pi)*log(Im(ZetaZero(n))/(2*Pi*e)) + 11/8) - n + 1). a(n) ~ (1 - sign(Im(zeta(1/2 + i*2*Pi*e*exp(LambertW((n - 11/8)/e))))))/2 where i = sqrt(-1). a(n) ~ floor(2*(RiemannSiegelTheta(Im(ZetaZero(n)))/Pi - floor(RiemannSiegelTheta(Im(ZetaZero(n)))/Pi))). There is a way to compute a(n) without prior knowledge of the exact locations of the Riemann zeta zeros. Let: FrancaLeclair(n) = 2*Pi*e*exp(LambertW((n - 11/8)/e)), NumberOfZetaZeros(t) = RiemannSiegelTheta(t)/Pi + Im(log(zeta(1/2 + i*t)))/Pi where i = sqrt(-1), Then: a(n) = n - 1 - NumberOfZetaZeros(FrancaLeclair(n)). Conjecture: a(n) ~ (1 + sign(tan((-RiemannSiegelTheta(im(zetazero (n)))))))/2. MATHEMATICA FrancaLeClair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]; Table[(1 + Sign[Im[ZetaZero[n]] - FrancaLeClair[n]])/2, {n, 1, 90}] CROSSREFS Cf. A002410, A273061, A282896, A282897. Sequence in context: A189212 A147781 A327216 * A082446 A191156 A144611 Adjacent sequences: A288637 A288638 A288639 * A288641 A288642 A288643 KEYWORD nonn AUTHOR Mats Granvik, Jun 17 2017 STATUS approved

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Last modified December 4 23:46 EST 2022. Contains 358572 sequences. (Running on oeis4.)