A327546 Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.

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

1,2

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.

Links

Table of n, a(n) for n=1..41.

Example

   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

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

Crossrefs

Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543.

Sequence in context: A097939 A249565 A174201 * A181844 A162506 A328609

Adjacent sequences: A327543 A327544 A327545 * A327547 A327548 A327549

Keyword

nonn,more

Author

Artur Jasinski, Sep 16 2019

Status

approved