Zeta iterations

[Gottfried]

Consider the (real) attractive fixed point of zeta z = -0.29590500557... In A069857, the author claims that this is also the limit of zeta^[m](z) for complex values of z as m tends to be large (or perhaps infinity, not sure if it exists). Can we show that the limit exists and -0.29590500557 IS the limit?

Balarka
.

I don't have a final answer, but the following observation are perhaps helpful to find such an answer.
A fixpoint is attracting, if the absolute value of its derivative is smaller than 1. So to see, whether z is attracting also for the complex plane its absolute value must be smaller in any direction of h in the derivative-formula (zeta(z+h/2)-zeta(z-h/2))/h .
To do this in Pari/GP h can be defined as h=exp(I*phi)*1e-20 for varying phi. For all phi tested the absolute of the derivative at z was smaller than 1. However - a heuristic is no proof...

Gottfried