A288640 a(n) = (1 + sign(Im(ZetaZero(n)) - 2*Pi*e*exp(LambertW((n - 11/8)/e))))/2.

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

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 * A361115 A082446 A191156

Adjacent sequences: A288637 A288638 A288639 * A288641 A288642 A288643

Keyword

nonn

Author

Mats Granvik, Jun 17 2017

Status

approved