A100060 a(n)=1 if the n-th second difference of the imaginary parts of the nontrivial zeros of the Riemann zeta function is positive, otherwise a(n)=0.

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1

Offset

1,1

Comments

Differences between zeta function gaps: increases are 1 and decreases are 0.

The ratios of the numbers of 0's to the number of 1's in the first 10^n differences are 0/1, 5/5, 50/50, 493/507, 4998/5002, 49949/50049, ...

Links

Table of n, a(n) for n=1..105.

J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown, On the distribution of gaps between zeros of the Zeta-Function, Quart. J. Math. oxford 36 (1985) 43-51.

A. M. Odlyzko, On the distribution of spacings between zeros of the Zeta Function, Math. Comp. 48 (177) (1987) 273-308.

A. M. Odlyzko, Tables

Index entries for zeta function.

Example

The first few positive t values of the zeros 1/2+i*t are (14.13..., 21.02..., 25.01..., 30.42..., 32.93..., 37.58..., 40.91..., 43.32...).
First differences are (6.88..., 3.98..., 5.41..., 2.51..., 4.65..., 3.33..., 2.40...).
Second differences are (-2.89..., 1.42..., -2.90..., 2.14..., -1.31..., -0.92...) which yields (0, 1, 0, 1, 0, 0, ...).

Mathematica

zz = { (* the list of values in the link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; Join[{1}, Table[ If[ yy[[n + 1]] > yy[[n]], 1, 0], {n, 104}]] (* Or *)
zz = { (* the list of values in the link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; xx = Drop[yy, 1] - Drop[yy, -1]; Join[{1}, Table[ If[ xx[[n]] > 0, 1, 0], {n, 104}]] (* Robert G. Wilson v, Jan 14 2005 *)
Flatten[{1, (Sign[Differences[Differences[Im[ZetaZero[Range[106]]]]]] + 1)/2}] (* Mats Granvik, Jul 23 2015 *)

Crossrefs

Cf. A102522, A102523.

Sequence in context: A267513 A004569 A361116 * A328101 A147850 A286046

Adjacent sequences: A100057 A100058 A100059 * A100061 A100062 A100063

Keyword

nonn

Author

Gary W. Adamson, Oct 31 2004

Extensions

Corrected and extended by Robert G. Wilson v, Jan 13 2005

Edited by M. F. Hasler, Jul 27 2015

Status

approved