I have just started learning about $L$-functions. The paper 'A History of the Prime Number Theorem' by Goldstein states that Dirichlet $L$-function $L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ is continuous and absolutely convergent for $Re(s)>1$, where $s \in \mathbb{C}$. Whilst absolute convergence is easy to prove, I have difficulty in proving the continuity, since $\chi$ takes varied values for a fixed number, say $k$.
For continuity, for a given $\delta>0$, I tried proving $|L(s_1,\chi)-L(s_2,\chi)|<\epsilon$ for some $\epsilon>0$ given that $|s_1-s_2|<\delta$, but I need assistance to deal with the difference of series.
You just have to show that the sum is uniformly convergent in a neighborhood of any $s$ with $\operatorname{Re}(s)>1$, since a uniform limit of continuous functions is continuous. This is easy: since $|n^s|=n^{\operatorname{Re}(s)}$, for any fixed $t>1$ the sum converges uniformly on the set where $\operatorname{Re}(s)>t$ (since the absolute values of the terms are bounded by their values when $s=t$ and you know the sum converges when $s=t$).