A069857 Decimal expansion of -C, where C = -0.2959050055752... is the real solution < 0 to zeta(x) = x.

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

Offset: 0

Author

Benoit Cloitre, Apr 27 2002

Keywords

• cons
• easy
• nonn

Comments

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....

C is not only a real fixed point of zeta, but the only attractive fixed point of Riemann zeta on the real line. - Balarka Sen, Feb 21 2013

Examples

			Let z=3+5I after 30 iterations : zeta_30(z)=-0.29590556499...-0.00000041029065...*I
		

Links

• Balarka Sen, Table of n, a(n) for n = 0..200

Programs

Mathematica

FindRoot[Zeta[z] - z, {z, 0}, WorkingPrecision -> 500] (* Balarka Sen, Feb 21 2013 *)

PARI

-solve(x=-1, 0, zeta(x)-x) \\ Michel Marcus, May 05 2020