Divergent Series as a topic of research

In some respects the theory of divergent series is still a very important part of number theory.

A large part of number theory concerns the study of Dirichlet series

$$f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$$ for some $a_n \in \mathbb{C}$ and some complex parameter $s \in \mathbb{C}$. Provided the $a_n$ satisfy some mild growth conditions, this series is absolutely convergent in some half-plane $\mathrm{Re}(s) > \sigma_0$.

One then wants to try to analytically continue this Dirichlet series to a meromorphic function on $\mathbb{C}$ and understand its zeros and poles. Analytic continuation replaces the classical treatment of divergent series by something more rigorous.

Important cases where one has an analytic continuation are for the Riemann zeta function and Dirichlet $L$-functions. Studying the analytic properties of Dirichlet series coming from Galois representations and automorphic forms is a very active area of research (the Langlands program).