A329742 -id:A329742 - cp's OEIS frontend

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))(log(z(m)/(2Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

1, 2, 4, 9, 13, 27, 34, 135, 159, 186, 212, 315, 363, 453, 693, 922, 1496, 4765, 6709, 44555, 73997, 82552, 87761, 95248, 415587, 420891, 1115578, 8546950, 24360732, 41820581, 1048449114, 3570918901, 35016977796

Offset: 1

Artur Jasinski, Jan 03 2020

nonn

No more records up to k = 103800788359.
Indices of zeros for successive maximal records of the normalized delta see A329742.
a(28)-a(33) computed by David Platt (2020).
Conjectural next terms: 1217992279429, 4088664936219.

			   n |  a(n) |    d(n)
  ---+-------+------------
   1 |     1 | 0.88871193
   2 |     2 | 0.76669277
   3 |     4 | 0.63017799
   4 |     9 | 0.57239954
   5 |    13 | 0.53062398
   6 |    27 | 0.52634271
   7 |    34 | 0.38628922
   8 |   135 | 0.37238098
   9 |   159 | 0.35780768
  10 |   186 | 0.32438582
  11 |   212 | 0.29105188
  12 |   315 | 0.24707528
  13 |   363 | 0.24343744
  14 |   453 | 0.23631515
  15 |   693 | 0.18028720
  16 |   922 | 0.13762601
  17 |  1496 | 0.08925253
  18 |  4765 | 0.04628960
  19 |  6709 | 0.04209838
  20 | 44555 | 0.04074628

Xavier Gourdon, The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004.
David Platt, Results computation of the smallest relative gaps between successive zeta zeros, 2020.

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

prec = 30; min = 10; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])], prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));
If[kk

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

cp's OEIS Frontend

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))(log(z(m)/(2Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

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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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cp's OEIS Frontend

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))*(log(z(m)/(2*Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

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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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Mathematica

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))(log(z(m)/(2Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.