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Successive negative minima of Gram's points g(n) of the Riemann zeta function.
5

%I #24 Apr 16 2022 15:04:00

%S 126,134,777,1165,2808,3782,12174,14374,23149,60780,117807,126085

%N Successive negative minima of Gram's points g(n) of the Riemann zeta function.

%C This sequence is subset of A114856.

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

%C For very small values of Gram's points, the distance between nearest zero of Riemann zeta function is very small.

%C For successive positive minima of Gram's points g(n) of the Riemann zeta function see A326890.

%H M. A. Korolev, <a href="https://doi.org/10.4213/sm8253">On small values of the Riemann zeta-function at Gram points</a>, Mat. Sb., 2014, Volume 205, Number 1, 67-86. In Russian.

%e n | a(n) | g(a(n)) = Zeta value

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

%e 1 | 126 | -0.02762949885719994

%e 2 | 134 | -0.01690039090339079

%e 3 | 777 | -0.00964626429746985

%e 4 | 1165 | -0.008575843736423

%e 5 | 2808 | -0.005747300941326

%e 6 | 3782 | -0.000760294730822

%e 7 | 12174 | -0.00045763304501

%e 8 | 14374 | -0.00027891005688

%e 9 | 23149 | -0.00007068683846

%e 10 | 60780 | -0.0000398945276

%e 11 | 117807 | -0.0000229487717

%e 12 | 126085 | -0.0000077126884

%t ee = 10; cc = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[ RiemannSiegelTheta][n Pi], 10]]];If[(kk < 0) && (Abs[kk] < ee), AppendTo[cc, n]; ee = Abs[kk]], {n, 1, 1000000}]; aa

%Y Cf. A114856, A326890.

%K nonn,more

%O 1,1

%A _Artur Jasinski_, Sep 13 2019