A114857 Decimal expansion of 0th Gram point.

1, 7, 8, 4, 5, 5, 9, 9, 5, 4, 0, 4, 1, 0, 8, 6, 0, 8, 1, 6, 8, 2, 6, 3, 3, 8, 4, 1, 2, 5, 1, 9, 0, 9, 7, 0, 3, 5, 6, 9, 3, 2, 8, 7, 4, 3, 3, 6, 9, 6, 4, 5, 2, 3, 9, 2, 1, 1, 8, 1, 1, 4, 8, 5, 9, 4, 8, 1, 6, 8, 7, 0, 0, 9, 2, 0, 1, 6, 0, 9, 5, 2, 1, 1, 7, 5, 1, 3, 4, 0, 4, 0, 8, 4, 8, 8, 2, 0, 8, 6, 7, 6

Offset: 2

Author

Eric W. Weisstein, Jan 02 2006

Keywords

• nonn
• cons

Examples

			17.8455995...
		

Links

• Eric Weisstein's World of Mathematics, Gram Point

Crossrefs

Cf. A114856, A114858.

Programs

Mathematica

First[ RealDigits[t /. FindRoot[ RiemannSiegelTheta[t] == 0, {t, 17}, WorkingPrecision -> 120], 10, 102]] (* Jean-François Alcover, Jun 07 2012 *)

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)
gram(0) \\ Charles R Greathouse IV, Jan 22 2022

PARI

solve(t=17.8,18,4*Pi+arg(gamma(1/4+I*t/2))-log(Pi)*t/2) \\ Charles R Greathouse IV, Mar 27 2023