

A114856


Indices n of Gram points g(n) for which (1)^n Z(g(n)) < 0, where Z(t) is the RiemannSiegel Zfunction.


13



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

1,1


REFERENCES

E. C. Titchmarsh, On van der Corput's Method and the zetafunction of Riemann IV, Quarterly Journal of Mathematics os5 (1934), pp. 98105.


LINKS

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


FORMULA

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


EXAMPLE

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


MATHEMATICA

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]] (* JeanFrançois Alcover, Oct 17 2012, after Eric W. Weisstein *)


CROSSREFS

Cf. A114857, A114858, A216700.
Sequence in context: A080539 A045167 A216063 * A165019 A025388 A025389
Adjacent sequences: A114853 A114854 A114855 * A114857 A114858 A114859


KEYWORD

nonn


AUTHOR

Eric W. Weisstein, Jan 02 2006


EXTENSIONS

Definition corrected by David Baugh, Apr 02 2008


STATUS

approved



