A114856 -id:A114856 - cp's OEIS frontend

A231157 Number of Gram blocks [g(j), g(j+1)) up to 10^n with 0 <= j < 10^n.

1, 10, 100, 916, 8374, 78694, 755132, 7297808, 71004697

Offset: 0

Arkadiusz Wesolowski, Nov 04 2013

nonn

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A231158-A231165.

A231165 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly three zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

Original entry on oeis.org

0, 6, 86, 1289, 14932, 166570

Offset: 3

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions

Cf. A114856, A231157-A231164.

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

A326891 Successive negative minima of Gram's points g(n) of the Riemann zeta function.

Original entry on oeis.org

126, 134, 777, 1165, 2808, 3782, 12174, 14374, 23149, 60780, 117807, 126085

Offset: 1

Artur Jasinski, Sep 13 2019

This sequence is subset of A114856.
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 positive minima of Gram's points g(n) of the Riemann zeta function see A326890.

			   n |  a(n)  | g(a(n)) = Zeta value
  ---+--------+---------------------
   1 |    126 | -0.02762949885719994
   2 |    134 | -0.01690039090339079
   3 |    777 | -0.00964626429746985
   4 |   1165 | -0.008575843736423
   5 |   2808 | -0.005747300941326
   6 |   3782 | -0.000760294730822
   7 |  12174 | -0.00045763304501
   8 |  14374 | -0.00027891005688
   9 |  23149 | -0.00007068683846
  10 |  60780 | -0.0000398945276
  11 | 117807 | -0.0000229487717
  12 | 126085 | -0.0000077126884

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.

Cf. A114856, A326890.

ee = 10; cc = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[ RiemannSiegelTheta][n Pi], 10]]];If[(kk < 0) && (Abs[kk] < ee), AppendTo[cc, n]; ee = Abs[kk]], {n, 1, 1000000}]; 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

A114857 Decimal expansion of 0th Gram point.

Original entry on oeis.org

1, 7, 8, 4, 5, 5, 9, 9, 5, 4, 0, 4, 1, 0, 8, 6, 0, 8, 1, 6, 8, 2, 6, 3, 3, 8, 4, 1, 2, 5, 1, 9, 0, 9, 7, 0, 3, 5, 6, 9, 3, 2, 8, 7, 4, 3, 3, 6, 9, 6, 4, 5, 2, 3, 9, 2, 1, 1, 8, 1, 1, 4, 8, 5, 9, 4, 8, 1, 6, 8, 7, 0, 0, 9, 2, 0, 1, 6, 0, 9, 5, 2, 1, 1, 7, 5, 1, 3, 4, 0, 4, 0, 8, 4, 8, 8, 2, 0, 8, 6, 7, 6

Offset: 2

Eric W. Weisstein, Jan 02 2006

			17.8455995...

Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A114858.

First[ RealDigits[t /. FindRoot[ RiemannSiegelTheta[t] == 0, {t, 17}, WorkingPrecision -> 120], 10, 102]] (* Jean-François Alcover, Jun 07 2012 *)

g0(n)=2*Pi*exp(1+lambertw((8*n+1)/exp(1)/8)) \\ approximate location of gram(n)
th(t)=arg(gamma(1/4+I*t/2))-log(Pi)*t/2 \\ theta, but off by some integer multiple of 2*Pi
thapprox(t)=log(t/2/Pi)*t/2-t/2-Pi/8+1/48/t-1/5760/t^3
RStheta(t)=my(T=th(t)); (thapprox(t)-T)\/(2*Pi)*2*Pi+T
gram(n)=my(G=g0(n),k=n*Pi); solve(x=G-.003,G+1e-8,RStheta(x)-k)
gram(0) \\ Charles R Greathouse IV, Jan 22 2022

solve(t=17.8,18,4*Pi+arg(gamma(1/4+I*t/2))-log(Pi)*t/2) \\ Charles R Greathouse IV, Mar 27 2023

A114858 Decimal expansion of first Gram point.

Original entry on oeis.org

2, 3, 1, 7, 0, 2, 8, 2, 7, 0, 1, 2, 4, 6, 3, 0, 9, 2, 7, 8, 9, 9, 6, 6, 4, 3, 5, 3, 8, 3, 0, 1, 5, 3, 2, 0, 5, 1, 7, 4, 7, 0, 9, 8, 3, 2, 6, 8, 4, 1, 6, 4, 6, 9, 7, 0, 8, 3, 0, 0, 8, 8, 5, 1, 9, 0, 2, 2, 9, 6, 6, 0, 3, 1, 9, 9, 3, 6, 0, 9, 3, 9, 0, 3, 3, 1, 0, 5, 7, 7, 4, 8, 3, 4, 4, 6, 3, 9, 1, 6, 0, 4

Offset: 2

Eric W. Weisstein, Jan 02 2006

			23.1702827...

Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A114857.

First[ RealDigits[t /. FindRoot[ RiemannSiegelTheta[t] == Pi, {t, 23}, WorkingPrecision -> 120], 10, 102]] (* Jean-François Alcover, Jun 07 2012 *)

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

A231158 Number of Gram blocks [g(j), g(j+2)) up to 10^n with 0 <= j < 10^n.

Original entry on oeis.org

0, 42, 780, 9445, 100203, 1034545, 10493487

Offset: 2

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A231157, A231159-A231165.

cp's OEIS Frontend

A231157 Number of Gram blocks [g(j), g(j+1)) up to 10^n with 0 <= j < 10^n.

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A231165 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly three zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

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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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A326891 Successive negative minima of Gram's points g(n) of the Riemann zeta function.

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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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A114857 Decimal expansion of 0th Gram point.

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PARI

PARI

A114858 Decimal expansion of first Gram point.

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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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