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Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.
1

%I #21 Oct 23 2019 15:56:49

%S 1,3,6,12,23,31,39,62,124,181,211,254,377,703,869,1207,1443,1702,1933,

%T 2565,3968,4657,4803,5815,6618,8569,13879,15321,25461,44681,58716,

%U 62728,68865,74399,83452,100050,167369,181802,185011,220569,259499

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

%C j-points occur when the real part of Riemann zeta function is zero but the imaginary part isn't zero.

%C The n-th j-point occur when Riemann-Siegel theta function is equal to Pi*(2n+1)/2.

%e n | a(n) | Zeta[1/2+I*j(a(n))] | j(a(n))

%e -----+--------+----------------------+------------

%e 1 | 1 | 0.6888099353665862*i | 25.49150821

%e 2 | 3 | 1.0716782759460156*i | 33.62379307

%e 3 | 6 | 1.3843203337013829*i | 43.99352729

%e 4 | 12 | 2.0558319047400831*i | 61.73354345

%e 5 | 23 | 2.2103659566253039*i | 89.57355850

%e 6 | 31 | 2.4259114706957412*i | 107.8332676

%e 7 | 39 | 2.5797839609135738*i | 125.0556067

%e 8 | 62 | 3.5676523298409918*i | 170.8597635

%e 9 | 124 | 3.9817183542258544*i | 279.9753243

%e 10 | 181 | 4.4992991376133266*i | 370.7853980

%e 11 | 211 | 4.7024313606767908*i | 416.3507516

%e 12 | 254 | 4.9763959256849833*i | 479.6816189

%e 13 | 377 | 6.0255895622763492*i | 651.5679685

%e 14 | 703 | 6.6869029304615494*i | 1068.801198

%e 15 | 869 | 6.9619624520146889*i | 1268.439833

%e 16 | 1207 | 7.0560068592571360*i | 1658.281364

%t 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

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

%K nonn,more

%O 1,2

%A _Artur Jasinski_, Sep 16 2019