A069857 - OEIS

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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

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OFFSET

0,1

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

LINKS

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Sequence in context: A339925 A171052 A021342 * A076841 A213819 A361013

Adjacent sequences: A069854 A069855 A069856 * A069858 A069859 A069860

KEYWORD

cons,easy,nonn

AUTHOR

Benoit Cloitre, Apr 27 2002

STATUS

approved