A114856 Indices n of ("bad") Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function.
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
Offset: 1
Keywords
Examples
(-1)^126 Z(g(126)) = -0.0276294988571999.... - _David Baugh_, Apr 02 2008
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- Mathematics Stack Exchange, crow, Can we prove B(n)=(1/4)G(n-1)G(n) is an indicator function ...
- 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.
- Timothy Trudgian, On the success and failure of Gram's Law and the Rosser Rule, Acta Arithmetica, 2011 | 148 | 3 | 225-256.
- Eric Weisstein's World of Mathematics, Gram Point.
Programs
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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]] (* Jean-François Alcover, Oct 17 2012, after Eric W. Weisstein *) -
PARI
g0(n)=2*Pi*exp(1+lambertw((8*n+1)/exp(1)/8)) \\ approximate location of gram(n) th(t)=arg(gamma(1/4+I*t/2))-log(Pi)*t/2 \\ theta, but off by some integer multiple of 2*Pi thapprox(t)=log(t/2/Pi)*t/2-t/2-Pi/8+1/48/t-1/5760/t^3 RStheta(t)=my(T=th(t)); (thapprox(t)-T)\/(2*Pi)*2*Pi+T gram(n)=my(G=g0(n), k=n*Pi); solve(x=G-.003, G+1e-8, RStheta(x)-k) Z(t)=exp(th(t)*I)*zeta(1/2+I*t) is(n)=my(G=gram(n));real((-1)^n*Z(G))<0 \\ Charles R Greathouse IV, Jan 22 2022
Formula
Trudgian shows that a(n) = O(n), that is, there exists some k such that a(n) <= k*n. - Charles R Greathouse IV, Aug 29 2012
In fact Trudgian shows that a(n) ≍ n, and further, there exist constants 1 < b < c such that b*n < a(n) < c*n. (See the paper's discussion of the Weak Gram Law.) - Charles R Greathouse IV, Mar 28 2023
Extensions
Definition corrected by David Baugh, Apr 02 2008