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.

1, 3, 5, 8, 14, 25, 33, 64, 79, 105, 126, 183, 256, 379, 567, 705, 795, 964, 1113, 1487, 1545, 1935, 2567, 3296, 3472, 3970, 6398, 6620, 8374, 8571, 9179, 10173, 10942, 11567, 13298, 13881, 15323, 25463, 28119, 36719, 64415, 70856, 83454, 100052, 103715, 146919, 185013, 220571, 399427, 491515, 516200, 857873, 880694, 1493008, 1613442

Offset

1,2

Comments

Between the n-th and (n+1)-th nontrivial Riemann zeros there is exactly one extremum of the Riemann-Siegel Z function.

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.

Successive records occur when gaps between two successive zeros are large.

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).

Links

Table of n, a(n) for n=1..55.

Tadej Kotnik, Computational estimation of the order of zeta(1/2 + i t), Mathematics of Computation, Vol. 73, No. 246 (2004), pp. 949-956.

Example

    n | a(n) |  max Z(t)  |     t
   ---+------+------------+------------
    1 |   1  |   2.340551 |  17.882582
    2 |   3  |   2.847472 |  27.735883
    3 |   5  |   2.942394 |  35.392730
    4 |   8  |  -3.664836 |  45.636113
    5 |  14  |  -4.166936 |  63.060427
    6 |  25  |   4.477140 |  90.723857
    7 |  33  |   5.193289 | 108.986790
    8 |  64  |  -5.980169 | 171.759106
    9 |  79  |   6.062599 | 199.651794

Mathematica

aa = {}; prec = 50; d = 30; e = 1/10^d; max = 0; Do[
p = N[Im[ZetaZero[t]], prec]; k = N[Im[ZetaZero[t + 1]], prec];
f = N[RiemannSiegelZ[(p + k)/2], prec];
g = N[RiemannSiegelZ[(p + k)/2 + e], prec];
Do[If[Abs[f - g] < 10^-40, Break[]];
  If[f < g, p = (p + k)/2 + e; f = N[RiemannSiegelZ[(p + k)/2], prec];
    g = N[RiemannSiegelZ[(p + k)/2 + e], prec], k = (p + k)/2;
   f = N[RiemannSiegelZ[(p + k)/2], prec];
   g = N[RiemannSiegelZ[(p + k)/2 + e], prec]], {m, 1, 1000}];
If[Abs[g] > max, max = Abs[g]; AppendTo[aa, t]], {t, 1, 1000}]; aa

Crossrefs

Cf. A114856, A254297, A255739, A255742, A326502, A329742.

Sequence in context: A109022 A023596 A208667 * A329742 A078065 A175378

Adjacent sequences: A329820 A329821 A329822 * A329824 A329825 A329826

Keyword

nonn

Author

Artur Jasinski, Nov 22 2019

Status

approved