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%I #34 Jan 31 2023 13:53:03
%S 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,
%T 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,
%U 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
%N 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.
%C Differences between zeta function gaps: increases are 1 and decreases are 0.
%C 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, ...
%H J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown, <a href="https://web.archive.org/web/20180821092113/http://www.math.sjsu.edu/~goldston/article06.PDF">On the distribution of gaps between zeros of the Zeta-Function</a>, Quart. J. Math. oxford 36 (1985) 43-51.
%H A. M. Odlyzko, <a href="http://dx.doi.org/10.1090/S0025-5718-1987-0866115-0">On the distribution of spacings between zeros of the Zeta Function</a>, Math. Comp. 48 (177) (1987) 273-308.
%H A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1">Tables</a>
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%e 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...).
%e First differences are (6.88..., 3.98..., 5.41..., 2.51..., 4.65..., 3.33..., 2.40...).
%e Second differences are (-2.89..., 1.42..., -2.90..., 2.14..., -1.31..., -0.92...) which yields (0, 1, 0, 1, 0, 0, ...).
%t 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 *)
%t 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 *)
%t Flatten[{1, (Sign[Differences[Differences[Im[ZetaZero[Range[106]]]]]] + 1)/2}] (* _Mats Granvik_, Jul 23 2015 *)
%Y Cf. A102522, A102523.
%K nonn
%O 1,1
%A _Gary W. Adamson_, Oct 31 2004
%E Corrected and extended by _Robert G. Wilson v_, Jan 13 2005
%E Edited by _M. F. Hasler_, Jul 27 2015
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