the " average restart " of a number.

[tommy1729]

Let z be a nonreal complex number.

z=z1
z2=exp(z1)
z3=exp(z2)
etc

Define f(x) as f = 1 if x is within the unit circle on the complex plane ( |x| < 1 ) and 0 otherwise.

Let ar_b(z) denote the " average restart " of z.
"ar_b" is defined by a limit of an integer n going to +oo.

ar_b(z) = lim_n [f(z1) + f(z2) + f(z3) + ... + f(zn)]/n^b = constant(z).

(b > 0)

What is known about this ??

regards

tommy1729