Let $f(y)\in \Bbb Z_p[y_0,y_1,....,y_n]$ be a homogeneous polynomial. Let $N_s$ be the number of zeros of $f$ in $\Bbb P^n(F_{p^s})$. Here, $\Bbb P^n(F_{p^s})$ denotes the $n$-th projective space defined over the finite field with $p^s$ elements.
My question is how does the series $\sum_{s=1}^{\infty}\frac{N_su^s}{s}$ define an analytic function on the disc $\{u\in \Bbb C:|u|<q^{-n}\}$.
You can get the radius of convergence for your power series by applying the root test which says that the radius of convergence is
$$ r=1/(\limsup_{s\to \infty} \sqrt[s]{N_s/s})$$
You could use the estimate that $N_s \le \#\mathbf P^n(\mathbf F_{p^s}) \le (n+1) (p^s)^n$.