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A325932 Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.

%I A325932 #67 Feb 16 2025 08:33:58
%S A325932 126,211,288,377,703,869,964,1933,1935,2675,3970,4265,4657,5225,6618,
%T A325932 8374,8569,18014,25461,28812,36719,50512,74399,83452,90051,103715,
%U A325932 146919,164189,185011,206716
%N A325932 Indices k of Gram points g(k) for successive negative maximal values of the Riemann zeta function on the critical line.
%C A325932 This sequence is subset of A114856.
%C A325932 The n-th Gram point occurs when the Riemann-Siegel theta function is equal to Pi*n.
%C A325932 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 A325932 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 A325932 For indices of successive positive minima of the Riemann zeta function at Gram points g(n) see A326890.
%C A325932 For indices of successive positive maxima of the Riemann zeta function at Gram points g(n) see A327543.
%C A325932 Computed record value of this sequence is a(n)=2601005843707 with value zeta[1/2+I*g(a(n))]= -119.630432107724 (Kotnik 2003).
%H A325932 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 A325932 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GramPoint.html">Gram Point</a>.
%e A325932    n |  a(n)  | Zeta[1/2+I*g(a(n))]  |    g(a(n))
%e A325932 -=---+--------+----------------------+------------
%e A325932    1 |    126 | -0.02762949885719994 |  282.4547208
%e A325932    2 |    211 | -0.38288957164454790 |  415.6014600
%e A325932    3 |    288 | -0.66545881605404208 |  527.6973416
%e A325932    4 |    377 | -0.83760106086093435 |  650.8910448
%e A325932    5 |    703 | -1.00455040613260376 | 1068.189532
%e A325932    6 |    869 | -1.27120822682165464 | 1267.847910
%e A325932    7 |    964 | -1.392200186869156   | 1379.419269
%e A325932    8 |   1933 | -1.413878403700959   | 2446.574386
%e A325932    9 |   1935 | -1.881639907182627   | 2448.681071
%e A325932   10 |   2675 | -2.062586314581326   | 3210.042865
%e A325932   11 |   3970 | -2.1482691132271     | 4479.035743
%e A325932   12 |   4265 | -2.1659698746279     | 4759.875045
%e A325932   13 |   4657 | -2.2554659693900     | 5129.256083
%e A325932   14 |   5225 | -2.4955901590107     | 5657.609720
%e A325932   15 |   6618 | -2.60670539564937    | 6924.738490
%e A325932   16 |   8374 | -2.95430731615046    | 8476.646123
%t A325932 ff = 0; aa = {}; Do[kk = Re[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][n Pi], 10]]];
%t A325932 If[kk < ff, AppendTo[aa, n]; ff = kk], {n, 1, 450000}]; aa
%Y A325932 Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543.
%K A325932 nonn
%O A325932 1,1
%A A325932 _Artur Jasinski_, Sep 16 2019