Zeta iterations

[Gottfried]

I see, but how did you derived that? Is it very straightforward from the definition of attractive fixed point but I cannot see it?

Hmm, perhaps you'll find better/more complete/more helpful explanations when googling for "basins of attraction" or "petals of attraction". Did you try our hyperoperation-wiki already?
Perhaps it is also helpful to remember, that the function zeta1(s) = zeta(s)+1/(1-s) is entire (and has a power series which is easily derivable with Pari/GP). Then also the derivative must be entire; and also the derivative of the remaing part zeta2(s) = -1/(1-s) is zeta2(s)'=1/(1-s)^2 which is finite only except at s=1.

But well, I think I'd stop here and confirm my much freehanded assumtions myself now...

Gottfried