A255742 -id:A255742 - cp's OEIS frontend

A255739 Indices of nontrivial zeros of Riemann zeta function whose imaginary part sets a record for the absolute minimal difference from an integer.

1, 2, 3, 9, 51, 473, 3233, 7657, 7722, 20002, 124170, 126137, 977155

Offset: 1

Omar E. Pol, Mar 17 2015

We consider here the imaginary part of 1/2 + iy = z, for which Zeta(z) is a zero.
No more terms below 600000. - Robert G. Wilson v, Sep 30 2015
Is there an Im(rho_k) that is also a positive integer? Is there a minimum gap between an Im(rho_k) and a positive integer? At present it is not known whether this sequence is finite or infinite. - Omar E. Pol, Oct 13 2015
No more terms below 2001052. - Amiram Eldar, Aug 10 2023

			-------------------------------------------------------------------
                                     Absolute      New
k      Im(rho_k)       A002410(k)   difference   record   n   a(n)
-------------------------------------------------------------------
1    14.134725142    >    14        0.134725142    Yes    1    1
2    21.022039639    >    21        0.022039639    Yes    2    2
3    25.010857580    >    25        0.010857580    Yes    3    3
4    30.424876126    >    30        0.424876126    Not
5    32.935061588    <    33        0.064938412    Not
6    37.586178159    <    38        0.413821841    Not
7    40.918719012    <    41        0.081280988    Not
8    43.327073281    >    43        0.327073281    Not
9    48.005150881    >    48        0.005150881    Yes    4    9
10   49.773832478    <    50        0.226167522    Not
...
where rho_k is the k-th nontrivial zero of Riemann zeta function.
We computed more digits of Im(rho_k), but in the above table only 9 digits beyond the decimal point appear.

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function.
Andrew M. Odlyzko, On the distribution of spacings between zeros of the zeta function.
Index entries for zeta function.

Cf. A002410, A255742.

mn = Infinity; k = 1; lst = {}; While[k < 2501, a = N[ Abs[ Im[ ZetaZero[
k]] - Round[ Im[ ZetaZero[ k]] ]], 32]; If[a < mn, AppendTo[lst, k];
Print[k]; mn = a]; k++]; lst (* Robert G. Wilson v, Sep 29 2015 *)

A255742(n) = A002410(a(n)).

a(6)-a(10) from Robert G. Wilson v, Sep 29 2015
a(11)-a(12) from Robert G. Wilson v, Sep 30 2015
a(13) using Odlyzko's tables added by Amiram Eldar, Aug 10 2023

A326890 Successive positive minima of Gram's points g(n) of the Riemann zeta function.

Original entry on oeis.org

1, 3, 8, 12, 26, 33, 62, 899, 1288, 3382, 3803, 17161, 97280, 208678, 368382, 45898152, 55785549, 65463721

Offset: 1

Artur Jasinski, Sep 13 2019

Gram's points occur when the imaginary part of Riemann zeta function is zero but the real part isn't zero.
For very small values of Gram's points the distance between nearest zero of Riemann zeta function is very small.
For successive negative minima of Gram's points g(n) of the Riemann zeta function see A326891.
a(16)-a(18) follow Korolev 2014.

			   n |  a(n)  | g(a(n)) = Zeta value
  ---+--------+---------------------
   1 |      1 | 1.457427047874012250
   2 |      3 | 0.925264643315366642
   3 |      8 | 0.688292371691853238
   4 |     12 | 0.538585793754601351
   5 |     26 | 0.491521463374527648
   6 |     33 | 0.14158237349601719
   7 |     62 | 0.00818833702586957
   8 |    899 | 0.00443821005886578
   9 |   1288 | 0.003877434204568
  10 |   3382 | 0.000203064538534
  11 |   3803 | 0.000137683252272
  12 |  17161 | 0.00011012022914
  13 |  97280 | 0.0000123785958
  14 | 208678 | 0.000010257478
  15 | 368382 | 0.0000000890976

M. A. Korolev, On small values of the Riemann zeta-function at Gram points, Mat. Sb., 2014, Volume 205, Number 1, 67-86. In Russian. In English.

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

ff = 10; aa = {}; Do[ kk = Re[Zeta[1/2 + I N[ InverseFunction[ RiemannSiegelTheta][n Pi], 10]]]; If[(kk > 0) && (kk < ff), AppendTo[aa, n]; ff = kk], {n, 1, 450000}]; aa

A325932 Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.

Original entry on oeis.org

126, 211, 288, 377, 703, 869, 964, 1933, 1935, 2675, 3970, 4265, 4657, 5225, 6618, 8374, 8569, 18014, 25461, 28812, 36719, 50512, 74399, 83452, 90051, 103715, 146919, 164189, 185011, 206716

Offset: 1

Artur Jasinski, Sep 16 2019

nonn

This sequence is subset of A114856.
The n-th Gram point occurs when the Riemann-Siegel theta function is equal to Pi*n.
Gram points occur when the imaginary part of the Riemann zeta function on the critical line is zero but the real part is nonzero.
For very small values of Riemann zeta function at Gram points, the distance to the nearest zero of Riemann zeta function is very small.
For indices of successive positive minima of the Riemann zeta function at Gram points g(n) see A326890.
For indices of successive positive maxima of the Riemann zeta function at Gram points g(n) see A327543.
Computed record value of this sequence is a(n)=2601005843707 with value zeta[1/2+I*g(a(n))]= -119.630432107724 (Kotnik 2003).

			   n |  a(n)  | Zeta[1/2+I*g(a(n))]  |    g(a(n))
-=---+--------+----------------------+------------
   1 |    126 | -0.02762949885719994 |  282.4547208
   2 |    211 | -0.38288957164454790 |  415.6014600
   3 |    288 | -0.66545881605404208 |  527.6973416
   4 |    377 | -0.83760106086093435 |  650.8910448
   5 |    703 | -1.00455040613260376 | 1068.189532
   6 |    869 | -1.27120822682165464 | 1267.847910
   7 |    964 | -1.392200186869156   | 1379.419269
   8 |   1933 | -1.413878403700959   | 2446.574386
   9 |   1935 | -1.881639907182627   | 2448.681071
  10 |   2675 | -2.062586314581326   | 3210.042865
  11 |   3970 | -2.1482691132271     | 4479.035743
  12 |   4265 | -2.1659698746279     | 4759.875045
  13 |   4657 | -2.2554659693900     | 5129.256083
  14 |   5225 | -2.4955901590107     | 5657.609720
  15 |   6618 | -2.60670539564937    | 6924.738490
  16 |   8374 | -2.95430731615046    | 8476.646123

T. Kotnik, Computational estimation of the order of zeta(1/2+it), Math. Comp. 73 (2004), 949-956.
Eric Weisstein's World of Mathematics, Gram Point.

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

ff = 0; aa = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][n Pi], 10]]];
If[kk < ff, AppendTo[aa, n]; ff = kk], {n, 1, 450000}]; aa

A327543 Indices n of Gram points g(n) for successive positive maxima of the Riemann zeta function on critical line.

Original entry on oeis.org

1, 2, 4, 7, 13, 24, 32, 63, 78, 125, 182, 255, 378, 566, 704, 794, 963, 1112, 1486, 1544, 1934, 2566, 3295, 3471, 3969, 6397, 6619, 8373, 8570, 9178, 10172, 10941, 11566, 12346, 13297, 13880, 15322, 25462, 28118, 36718, 64414, 70855, 83453, 100051, 103714, 146918, 185012, 220570

Offset: 1

Artur Jasinski, Sep 16 2019

nonn

Gram points occur when the imaginary part of Riemann zeta function is zero but the real part nonzero.
The n-th Gram point occurs when the Riemann-Siegel theta function is equal to Pi*n.
For indices of Gram points g(n) for successive positive minima of the Riemann zeta function on critical line see A326890.
For indices of Gram points g(n) for successive negative minima of the Riemann zeta function on critical line see A326891.
For indices of Gram points g(n) for successive negative maxima of the Riemann zeta function on critical line see A325932.

			   n | a(n) | Zeta(1/2 + I*g(a(n))) |    g(a(n))
  ---+------+-----------------------+------------
   1 |    1 |  1.45742704787401225  | 23.17028270
   2 |    2 |  2.84509123805192195  | 27.67018222
   3 |    4 |  2.93812153849374056  | 35.46718430
   4 |    7 |  3.66290294911991710  | 45.59302898
   5 |   13 |  4.16439875850106581  | 63.10186798
   6 |   24 |  4.47536695704548069  | 90.75295338
   7 |   32 |  5.18702282127077889  | 108.9364311
   8 |   63 |  5.97089319007464658  | 171.8101081
   9 |   78 |  6.06256772354879599  | 199.6489681
  10 |  125 |  7.00315163729736922  | 280.8024294
  11 |  182 |  7.56958843983997014  | 371.5556258
  12 |  255 |  8.24960849238073236  | 480.4061559
  13 |  378 |  9.14820901096157903  | 652.2447407
  14 |  566 |  9.37745383604127446  | 897.7841913
  15 |  704 |  9.81879930244819679  | 1069.412795
  16 |  794 | 10.35506137680061993  | 1178.447136

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

ff = 0; aa = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][n Pi], 10]]]; If[kk > ff, AppendTo[aa, n]; ff = kk], {n, 1, 250000}]; aa

A329742 Indices n of Riemann zeta zeros for successive records of the normalized delta defined as d(n) = (z(n+1)-z(n))*(log(z(n)/(2Pi))/(2Pi)) where z(n) is the imaginary part of the n-th Riemann zero.

Original entry on oeis.org

1, 3, 5, 8, 14, 25, 33, 64, 126, 213, 256, 379, 1704, 1935, 2292, 8571, 10942, 12347, 13298, 15323, 36719, 46589, 103715, 185013, 880694, 1493008, 3206674, 12534781, 14145077, 22653912, 24246374, 33742399, 65336924, 298466597, 566415148, 1938289664, 2122614029, 4020755339, 4219726754, 16265396008, 17003807756

Offset: 1

Artur Jasinski, Nov 20 2019

nonn

No more records up to n = 103800788359.
d(17003807756) = 4.3018209763411.
Successive records occur when gaps between two successive zeros are large.
Recent record of normalized delta computed by Hiary at 2011 occurs for n=436677148707320393224019748290912 where d(n) = 5.77979.
Conjectural next term: 77528045597.
Indices of zeros for successive minimal records of the normalized delta see A328656.

			   n |   a(n)  |  d(n)
  ---+---------+---------
   1 |       1 | 0.88871
   2 |       3 | 1.19034
   3 |       5 | 1.22634
   4 |       8 | 1.43763
   5 |      14 | 1.54672
   6 |      25 | 1.55244
   7 |      33 | 1.74300
   8 |      64 | 1.83656
   9 |     126 | 1.95400
  10 |     213 | 1.95626
  11 |     256 | 1.99205
  12 |     379 | 2.20138
  13 |    1704 | 2.20198
  14 |    1935 | 2.45843
  15 |    2292 | 2.46772
  16 |    8571 | 2.48347
  17 |   10942 | 2.50594
  18 |   12347 | 2.50648
  19 |   13298 | 2.52517
  20 |   15323 | 2.67728
  21 |   36719 | 2.76188
  22 |   46589 | 2.80523
  23 |  103715 | 2.83121
  24 |  185013 | 3.11058
  25 |  880694 | 3.21426
  26 | 1493008 | 3.30347

Artur Jasinski, Table of n, a(n) for n = 1..41
Ghaith Ayesh Hiary, Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891-946. MR 2831110 (2012g:11154).
David Platt, Results computation of the largest relative gaps between successive zeta zeros, 2020.

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

prec = 30; max = 0; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])],prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));
If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa

a(27)-a(41) computed by David Platt, Jan 03 2020

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.

Original entry on oeis.org

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

A327546 Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.

Original entry on oeis.org

1, 3, 6, 12, 23, 31, 39, 62, 124, 181, 211, 254, 377, 703, 869, 1207, 1443, 1702, 1933, 2565, 3968, 4657, 4803, 5815, 6618, 8569, 13879, 15321, 25461, 44681, 58716, 62728, 68865, 74399, 83452, 100050, 167369, 181802, 185011, 220569, 259499

Offset: 1

Artur Jasinski, Sep 16 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)  | Zeta[1/2+I*j(a(n))]  |  j(a(n))
-----+--------+----------------------+------------
   1 |      1 | 0.6888099353665862*i |  25.49150821
   2 |      3 | 1.0716782759460156*i |  33.62379307
   3 |      6 | 1.3843203337013829*i |  43.99352729
   4 |     12 | 2.0558319047400831*i |  61.73354345
   5 |     23 | 2.2103659566253039*i |  89.57355850
   6 |     31 | 2.4259114706957412*i |  107.8332676
   7 |     39 | 2.5797839609135738*i |  125.0556067
   8 |     62 | 3.5676523298409918*i |  170.8597635
   9 |    124 | 3.9817183542258544*i |  279.9753243
  10 |    181 | 4.4992991376133266*i |  370.7853980
  11 |    211 | 4.7024313606767908*i |  416.3507516
  12 |    254 | 4.9763959256849833*i |  479.6816189
  13 |    377 | 6.0255895622763492*i |  651.5679685
  14 |    703 | 6.6869029304615494*i | 1068.801198
  15 |    869 | 6.9619624520146889*i | 1268.439833
  16 |   1207 | 7.0560068592571360*i | 1658.281364

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

ff = 0; aa = {}; Do[kk = Im[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][(2 n + 1) Pi/2],10]]]; If[kk > ff, AppendTo[aa, n]; ff = kk], {n, 1, 100051}]; aa

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A255739 Indices of nontrivial zeros of Riemann zeta function whose imaginary part sets a record for the absolute minimal difference from an integer.

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A326890 Successive positive minima of Gram's points g(n) of the Riemann zeta function.

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A325932 Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.

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A327543 Indices n of Gram points g(n) for successive positive maxima of the Riemann zeta function on critical line.

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A329742 Indices n of Riemann zeta zeros for successive records of the normalized delta defined as d(n) = (z(n+1)-z(n))*(log(z(n)/(2Pi))/(2Pi)) where z(n) is the imaginary part of the n-th Riemann zero.

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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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A327546 Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.

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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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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.