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A114856 Indices n of Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function. 13


%S 126,134,195,211,232,254,288,367,377,379,397,400,461,507,518,529,567,

%T 578,595,618,626,637,654,668,692,694,703,715,728,766,777,793,795,807,

%U 819,848,857,869,887,964,992,995,1016,1028,1034,1043,1046,1071,1086

%N Indices n of Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function.

%D 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.

%H Timothy Trudgian, <a href="http://www.cs.uleth.ca/~trudgian/TrudgianActaArithmeticaGram.pdf">On the success and failure of Gram's Law and the Rosser Rule</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GramPoint.html">Gram Point</a>

%F 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

%e E.g. (-1)^126 Z(g(126)) = -0.0276294988571999... [David Baugh]

%t 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-Fran├žois Alcover_, Oct 17 2012, after _Eric W. Weisstein_ *)

%Y Cf. A114857, A114858, A216700.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jan 02 2006

%E Definition corrected by David Baugh, Apr 02 2008

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Last modified December 21 01:10 EST 2014. Contains 252291 sequences.