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

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

Author

Artur Jasinski, Sep 16 2019

Keywords

• nonn

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