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 1-4 of 4 results.

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 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.