Let $a, b \in (0, 1]$ and define $S_N:=\sum_{n=1}^N a^{-n} n^{-b}$.
Question
What is a good upper bound for $S_N$ ?
Observations
By a simple (and probably careless) application of Cauchy-Schwarz, one can already get
$$ S_N \le (\max_{n=1}^N a^{-n})\sum_{n=1}^N n^{-b} \le \frac{1}{a}\begin{cases}\frac{1}{1-b}(N^{1-b}-b),&\mbox{ if }0 \le b < 1\\ \log(N) + 1,&\mbox{ if }b=1. \end{cases} $$ See this post for details.
I was wondering whether one can do much better than this, and using what kind of tools.
Edit
One of the users have suggested trying to look around incomplete gamma functions. Indeed, according to wolfram alpha, $$\int_{1}^{N} a^{-t}t^{-b}dt = \log^{b-1}(a)\left(\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))\right), $$ where $\Gamma(1-a,x)$ defines an incomplete gamma function.
I'm wondering whether there are any upper bounds for such an expression.