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

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

Offset: 1

Author

Artur Jasinski, Sep 13 2019

Keywords

• nonn
• more

Comments

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.

Examples

			   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
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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