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Indices n of j-points j(n) for successive positive minima of the Riemann zeta function on critical line.
0

%I #7 Jan 02 2020 20:49:50

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

%N Indices n of j-points j(n) for successive positive minima 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) | j(a(n)) | zeta(1/2+i*j(a(n)))

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

%e 1 | 1 | 25.49150821 | 0.68880994 * i

%e 2 | 9 | 53.21405637 | 0.59984107 * i

%e 3 | 14 | 67.13274840 | 0.09483571 * i

%e 4 | 27 | 98.85689819 | 0.09031281 * i

%e 5 | 38 | 122.94885747 | 0.00316160 * i

%e 6 | 288 | 528.40629391 | 0.00013121 * i

%e 7 | 28171 | 24370.31450783 | 0.00004727 * i

%e 8 | 42680 | 35149.21796047 | 0.00000366 * i

%t 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

%Y Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543, A327546.

%K nonn,more

%O 1,2

%A _Artur Jasinski_, Nov 20 2019