A114856 -id:A114856 - cp's OEIS frontend

A329751 Indices n of j-points j(n) for successive positive minima of the Riemann zeta function on critical line.

1, 9, 14, 27, 38, 288, 28171, 42680

Offset: 1

Artur Jasinski, Nov 20 2019

j-points occur when the real part of Riemann zeta function is zero but the imaginary part isn't zero.

The n-th j-point occur when Riemann-Siegel theta function is equal to Pi*(2n+1)/2.

			   n |  a(n)  |   j(a(n))      | zeta(1/2+i*j(a(n)))
  ---+--------+----------------+----------------------
   1 |      1 |    25.49150821 | 0.68880994 * i
   2 |      9 |    53.21405637 | 0.59984107 * i
   3 |     14 |    67.13274840 | 0.09483571 * i
   4 |     27 |    98.85689819 | 0.09031281 * i
   5 |     38 |   122.94885747 | 0.00316160 * i
   6 |    288 |   528.40629391 | 0.00013121 * i
   7 |  28171 | 24370.31450783 | 0.00004727 * i
   8 |  42680 | 35149.21796047 | 0.00000366 * i

Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543, A327546.

prec=20;ff = 10; aa = {}; Do[kk = Im[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][(2 n + 1) Pi/2], prec]]]; If[(kk < ff) && (kk > 0), AppendTo[aa, n]; ff = kk], {n,  1, 50000}]; aa

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.

Original entry on oeis.org

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

Artur Jasinski, Nov 22 2019

nonn

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).

			    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

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

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

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

A331100 a(n) is the index of the first occurrence of exactly n zeta zeros in the interval between g(n) and g(n+1) Gram points.

Original entry on oeis.org

-1, 126, 2145, 368714779, 3680295786520

Offset: 1

Artur Jasinski, Jan 09 2020

a(4)-a(5) computed by Gourdon 2004.

a(6) > 10^23.

			The first nontrivial Riemann zero is situated between g(-1) and g(0) so a(1)=-1.

Xavier Gourdon, The 10^13 first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004.

Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A327543.

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A329751 Indices n of j-points j(n) for successive positive minima of the Riemann zeta function on critical line.

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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.

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A331100 a(n) is the index of the first occurrence of exactly n zeta zeros in the interval between g(n) and g(n+1) Gram points.

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