login
Continued fraction expansion of abs(C) where C=-0.2959050055752...is the real negative solution to zeta(x)=x.
0

%I #8 Jul 06 2024 10:25:20

%S 0,3,2,1,1,1,2,1,7,14,1,2,10,1,5,3,1,7,2,1,2,2,2,4,1,1,12,1,1,1,14,2,

%T 10,3,5,6,2,1,6,13,1,2,2,4,8,1,4,8,2,1,16,1,1,1,1,4,2,1,1,1,3,13,4,1,

%U 2,1,6,1,1,2,43,1,3,1,1,2,2,2,1,2,2,2,10,5,4,8,1,5,3,2,1,1,3,2,19

%N Continued fraction expansion of abs(C) where C=-0.2959050055752...is the real negative solution to zeta(x)=x.

%C Start from any complex number z=x+iy, not solution to zeta(z)=z, iterate the zeta function on z. If zeta_m(z) (=zeta(zeta(....(z)..)) m times) has a limit when m grows, then this limit seems to always be the real number : C=-0.2959050055752....Example: if z=3+5I after 30 iterations : zeta_30(z)=-0.29590556499...-0.00000041029065...*I

%o (PARI) \p150 contfrac(abs(solve(X=-1,0,zeta(X)-X)))

%Y Cf. A069857 (decimal expansion).

%K base,cofr,easy,nonn

%O 0,2

%A _Benoit Cloitre_, Jun 19 2002

%E Offset changed by _Andrew Howroyd_, Jul 06 2024