How to extend $f(z,b)$ and $h(z,b)$ to $|z|>1$?

Question

Let $b>1$.

Define $log_b^*(n)$ as taking the logaritm base $b$ as many times a possible such that $log_b^*(n) > 1$.

Now define $g(n) = ln_b(n) \ln_b^{[2]}(n) \ln_b^{[3]}(n) ... log_b^*(n)$

where $ln_b^{[k]}$ means taking log base $b$ a number of $k$ times.

Now let $z$ be a complex number and

$$f(z,b) = \sum_{n>b}^{\infty} g(n) z^n$$

$$h(z,b) = \sum_{n>b}^{\infty} \frac{ z^n}{g(n)}$$

What are the analytic continuations of $f(z,b)$ and $h(z,b)$ with respect $z$ ? Or other continuations if there is a natural boundary ?

So basically : How to extend $f(z,b)$ and $h(z,b)$ to $|z|>1$ ?

related, especially for understanding different $b$ :

And maybe this is handy too :

I was wondering if the Borel transform, Borel transform and summation etc is all we need ?

The convergence radius is probably $1$ in each case, so you can't continue in a neighbourhood of $|z| = 1$. Perhaps, you can continue them in some part of the plane but not all of it. — Cactus, Jun 12 '25 at 13:08
@Cactus I know the taylor radius is 1. A perhaps statement without motivation is not much tbh. Why do you believe that ? Are you familiar with analytic continuation or other continuations ? — mick, Jun 14 '25 at 11:01
turning $g(n)$ into a nice interpolation taylor series and then using polylog to sum it will probably work. — mick, Jun 16 '25 at 07:16

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