A282896 Indices n such that sign(Im(zetazero(n)) - 2*Pi*e*exp(LambertW((n - 11/8)/e))) = 1.

2, 4, 6, 7, 9, 12, 13, 15, 19, 22, 24, 26, 27, 30, 31, 34, 38, 39, 42, 43, 45, 47, 48, 51, 53, 56, 57, 61, 62, 63, 65, 66, 70, 71, 75, 77, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 95, 96, 97, 100, 101, 102, 106, 107

Offset: 1

Author

Mats Granvik, Feb 24 2017

Keywords

• nonn

Comments

The beginning of this sequence agrees with the sequence of numbers n such that floor(Im(zetazero(n))/(2*Pi)*log(Im(zetazero(n))/(2*Pi*e)) + 11/8 - n + 1) = 1, but disagrees later. The first disagreements are at n = 28813, 30264, 36720, 45925, 46590, 50513, 55258, 63925, 64573, 73615, 78374, 82247, 94463, ... and these numbers are in a(n) but not in the sequence that uses the floor function.

The beginning of this sequence also agrees with numbers n such that sign(Im(zeta(1/2 + I*2*Pi*e*exp(LambertW((n - 11/8)/e))))) = -1, but disagrees later. The first numbers that are in a(n) but not in the sequence that uses the sign function are n = 28814, 30265, 36721, 45926, 46591, ... The first numbers that are in the sequence that uses the sign function but not in a(n) are n = 39325, 44468, ... Compare this to the sequences in Remark 2 in A282897.

From Mats Granvik, Jun 17 2017: (Start)

There is at least an initial agreement between a(n) and the positions of 1 in: floor(2*(RiemannSiegelTheta(Im(ZetaZero(n)))/Pi - floor(RiemannSiegelTheta(Im(ZetaZero(n)))/Pi))).

There is at least an initial agreement between a(n) and the positions of 1 in the sequence computed without prior knowledge of the exact locations of the Riemann zeta zeros, that instead uses the Franca-Leclair asymptotic as the argument to the zeta zero counting function. See the Mathematica program below.

Complement to A282897.

(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5500

Crossrefs

Cf. A002505, A135297, A153815, A273061, A282793, A282794, A282896, A282897.

Programs

Mathematica

FrancaLeClair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]]; f = Table[Sign[ Im[ZetaZero[n]] - FrancaLeClair[n]], {n, 1, 110}]; Flatten[Position[f, 1]]

Formula

a(n) = positions where A288640 = 1.