cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Artur Jasinski, Sep 16 2019

Keywords

Comments

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).

Examples

			   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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Sep 13 2019

Keywords

Comments

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.

Examples

			   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

Links

Crossrefs

Cf. A114856, A326890.

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Sep 16 2019

Keywords

Comments

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.

Examples

			   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

Crossrefs

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

Programs

  • Mathematica
    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

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

Views

Author

Artur Jasinski, Sep 16 2019

Keywords

Comments

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.

Examples

			   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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Nov 20 2019

Keywords

Comments

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.

Examples

			   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

Crossrefs

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

Programs

  • Mathematica
    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

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

Views

Author

Artur Jasinski, Jan 09 2020

Keywords

Comments

a(4)-a(5) computed by Gourdon 2004.
a(6) > 10^23.

Examples

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

Links

Crossrefs

Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A327543.
Showing 1-6 of 6 results.

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