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A114856
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Indices n of Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function.
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3
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126, 134, 195, 211, 232, 254, 288, 367, 377, 379, 397, 400, 461, 507, 518, 529, 567, 578, 595, 618, 626, 637, 654, 668, 692, 694, 703, 715, 728, 766, 777, 793, 795, 807, 819, 848, 857, 869, 887, 964, 992, 995, 1016, 1028, 1034, 1043, 1046, 1071, 1086
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OFFSET
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1,1
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REFERENCES
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E. C. Titchmarsh, On van der Corput's Method and the zeta-function of Riemann IV, Quarterly Journal of Mathematics os-5 (1934), pp. 98-105.
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LINKS
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Table of n, a(n) for n=1..49.
Timothy Trudgian, On the success and failure of Gram's Law and the Rosser Rule
Eric Weisstein's World of Mathematics, Gram Point
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FORMULA
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Trudgian shows that a(n) = O(n), that is, there exists some k such that a(n) <= kn. - Charles R Greathouse IV, Aug 29 2012
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EXAMPLE
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E.g. (-1)^126 Z(g(126)) = -0.0276294988571999... [David Baugh]
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MATHEMATICA
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g[n_] := (g0 /. FindRoot[ RiemannSiegelTheta[g0] == Pi*n, {g0, 2*Pi*Exp[1 + ProductLog[(8*n + 1)/(8*E)]]}, WorkingPrecision -> 16]); Reap[For[n = 1, n < 1100, n++, If[(-1)^n*RiemannSiegelZ[g[n]] < 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 17 2012, after Eric W. Weisstein *)
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CROSSREFS
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Cf. A114857, A114858, A216700.
Sequence in context: A080539 A045167 A216063 * A165019 A025388 A025389
Adjacent sequences: A114853 A114854 A114855 * A114857 A114858 A114859
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein, Jan 02 2006
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EXTENSIONS
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Definition corrected by David Baugh, Apr 02 2008
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STATUS
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approved
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