I get confused reading about L-series and there is a lemma on infinite series. The question should only concern about analysis and there should be no number theory involved. The lemma is below,
Let $\{u_j\}$ be a sequence of real numbers, all $\geq 2$ and suppose
$$f(s)=\prod_{j=1}^{\infty}(1-u^{-s}_j)^{-1}$$
is uniformly convergent in $D(1, \delta, \epsilon):= \{s\in \mathbb{C}|Re(S)\geq 1+\delta, |arg(s-1)|\leq \pi/2-\epsilon\}$ for every positive $\delta$ and $\epsilon$. Then
$$\log f(s)=\sum_{j=1}^{\infty}u^{-s}_j+g(s)$$
where $g(s)$ is bounded in a neighborhood of $s=1$.
Then the proof starts with, uniform convergence allows us to manipulate
$$\log f(s)=-\sum_{j=1}^{\infty}\log(1-u_j^{-s})=\sum_{j=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{mu_j^{sm}}=\sum_{j=1}^{\infty}\frac{1}{u_j^s} + \sum_{j=1}^{\infty}\sum_{m=2}^{\infty}\frac{1}{mu_j^{sm}}$$
I think the last equality follows since the two series $\sum_{j=1}^{\infty}\frac{1}{u_j^s} $, $\sum_{j=1}^{\infty}\sum_{m=2}^{\infty}\frac{1}{mu_j^{sm}}$ converges absolutely. Take the absolute values and the series is increasing, thus converge as they are bounded by $\log f(s)$. Is that correct?
If my reasoning is correct, why we need UNIFORM convergence in this case? Thank you.