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.
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
KEYWORD
nonn,more
AUTHOR
Artur Jasinski, Sep 16 2019
STATUS
approved