login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

%I

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 23 19:50 EST 2014. Contains 249865 sequences.