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

Author

Gary W. Adamson, Oct 31 2004

Keywords

• nonn

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, ...

Examples

			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, ...).
		

Links

• 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.

Crossrefs

Cf. A102522, A102523.

Programs

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 *)

Extensions

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

Edited by M. F. Hasler, Jul 27 2015