A325932 Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.

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

Author

Artur Jasinski, Sep 16 2019

Keywords

• nonn

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

• 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.

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