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A329823 Indices n of Riemann zeta zeros where the Riemann-Siegel Z function sets successive records of maximum absolute values abs(Z(t)) in the interval between the n-th and (n+1)-th zeros.

%I A329823 #38 Jan 03 2020 13:16:14
%S A329823 1,3,5,8,14,25,33,64,79,105,126,183,256,379,567,705,795,964,1113,1487,
%T A329823 1545,1935,2567,3296,3472,3970,6398,6620,8374,8571,9179,10173,10942,
%U A329823 11567,13298,13881,15323,25463,28119,36719,64415,70856,83454,100052,103715,146919,185013,220571,399427,491515,516200,857873,880694,1493008,1613442
%N A329823 Indices n of Riemann zeta zeros where the Riemann-Siegel Z function sets successive records of maximum absolute values abs(Z(t)) in the interval between the n-th and (n+1)-th zeros.
%C A329823 Between the n-th and (n+1)-th nontrivial Riemann zeros there is exactly one extremum of the Riemann-Siegel Z function.
%C A329823 If n is odd then Z(t) > 0 else Z(t) < 0, where z(n) is the imaginary part of the n-th Riemann zero, z(n) < t < z(n+1), and Z'(t) = 0.
%C A329823 Successive records occur when gaps between two successive zeros are large.
%C A329823 This sequence has many of the same terms as A329742. But some terms in A329742 are absent from this sequence (e.g., 213, 1704, 2295), and this sequence includes some terms that are absent from A329742 (e.g., 79, 105, 183).
%H A329823 Tadej Kotnik, <a href="https://doi.org/10.1090/S0025-5718-03-01568-0">Computational estimation of the order of zeta(1/2 + i t)</a>, Mathematics of Computation, Vol. 73, No. 246 (2004), pp. 949-956.
%e A329823     n | a(n) |  max Z(t)  |     t
%e A329823    ---+------+------------+------------
%e A329823     1 |   1  |   2.340551 |  17.882582
%e A329823     2 |   3  |   2.847472 |  27.735883
%e A329823     3 |   5  |   2.942394 |  35.392730
%e A329823     4 |   8  |  -3.664836 |  45.636113
%e A329823     5 |  14  |  -4.166936 |  63.060427
%e A329823     6 |  25  |   4.477140 |  90.723857
%e A329823     7 |  33  |   5.193289 | 108.986790
%e A329823     8 |  64  |  -5.980169 | 171.759106
%e A329823     9 |  79  |   6.062599 | 199.651794
%t A329823 aa = {}; prec = 50; d = 30; e = 1/10^d; max = 0; Do[
%t A329823 p = N[Im[ZetaZero[t]], prec]; k = N[Im[ZetaZero[t + 1]], prec];
%t A329823 f = N[RiemannSiegelZ[(p + k)/2], prec];
%t A329823 g = N[RiemannSiegelZ[(p + k)/2 + e], prec];
%t A329823 Do[If[Abs[f - g] < 10^-40, Break[]];
%t A329823   If[f < g, p = (p + k)/2 + e; f = N[RiemannSiegelZ[(p + k)/2], prec];
%t A329823     g = N[RiemannSiegelZ[(p + k)/2 + e], prec], k = (p + k)/2;
%t A329823    f = N[RiemannSiegelZ[(p + k)/2], prec];
%t A329823    g = N[RiemannSiegelZ[(p + k)/2 + e], prec]], {m, 1, 1000}];
%t A329823 If[Abs[g] > max, max = Abs[g]; AppendTo[aa, t]], {t, 1, 1000}]; aa
%Y A329823 Cf. A114856, A254297, A255739, A255742, A326502, A329742.
%K A329823 nonn
%O A329823 1,2
%A A329823 _Artur Jasinski_, Nov 22 2019