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A329751
Indices n of j-points j(n) for successive positive minima of the Riemann zeta function on critical line.
0
1, 9, 14, 27, 38, 288, 28171, 42680
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) | 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
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
KEYWORD
nonn,more
AUTHOR
Artur Jasinski, Nov 20 2019
STATUS
approved