A329751 Indices n of j-points j(n) for successive positive minima of the Riemann zeta function on critical line.

1, 9, 14, 27, 38, 288, 28171, 42680

Offset: 1

Author

Artur Jasinski, Nov 20 2019

Keywords

• nonn
• more

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