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%I #66 Sep 29 2019 10:53:23
%S 126,211,288,377,703,869,964,1933,1935,2675,3970,4265,4657,5225,6618,
%T 8374,8569,18014,25461,28812,36719,50512,74399,83452,90051,103715,
%U 146919,164189,185011,206716
%N Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.
%C This sequence is subset of A114856.
%C The n-th Gram point occurs when the Riemann-Siegel theta function is equal to Pi*n.
%C Gram points occur when the imaginary part of the Riemann zeta function on the critical line is zero but the real part is nonzero.
%C For very small values of Riemann zeta function at Gram points, the distance to the nearest zero of Riemann zeta function is very small.
%C For indices of successive positive minima of the Riemann zeta function at Gram points g(n) see A326890.
%C For indices of successive positive maxima of the Riemann zeta function at Gram points g(n) see A327543.
%C Computed record value of this sequence is a(n)=2601005843707 with value zeta[1/2+I*g(a(n))]= -119.630432107724 (Kotnik 2003).
%H T. Kotnik, <a href="https://doi.org/10.1090/S0025-5718-03-01568-0">Computational estimation of the order of zeta(1/2+it)</a>, Math. Comp. 73 (2004), 949-956.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GramPoint.html">Gram Point</a>.
%e n | a(n) | Zeta[1/2+I*g(a(n))] | g(a(n))
%e -=---+--------+----------------------+------------
%e 1 | 126 | -0.02762949885719994 | 282.4547208
%e 2 | 211 | -0.38288957164454790 | 415.6014600
%e 3 | 288 | -0.66545881605404208 | 527.6973416
%e 4 | 377 | -0.83760106086093435 | 650.8910448
%e 5 | 703 | -1.00455040613260376 | 1068.189532
%e 6 | 869 | -1.27120822682165464 | 1267.847910
%e 7 | 964 | -1.392200186869156 | 1379.419269
%e 8 | 1933 | -1.413878403700959 | 2446.574386
%e 9 | 1935 | -1.881639907182627 | 2448.681071
%e 10 | 2675 | -2.062586314581326 | 3210.042865
%e 11 | 3970 | -2.1482691132271 | 4479.035743
%e 12 | 4265 | -2.1659698746279 | 4759.875045
%e 13 | 4657 | -2.2554659693900 | 5129.256083
%e 14 | 5225 | -2.4955901590107 | 5657.609720
%e 15 | 6618 | -2.60670539564937 | 6924.738490
%e 16 | 8374 | -2.95430731615046 | 8476.646123
%t ff = 0; aa = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][n Pi], 10]]];
%t If[kk < ff, AppendTo[aa, n]; ff = kk], {n, 1, 450000}]; aa
%Y Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543.
%K nonn
%O 1,1
%A _Artur Jasinski_, Sep 16 2019
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