About a year ago, while studying real analysis, I got very much interested in divergent series. I discussed possible research topics related to divergent series with my teachers but couldn't find any. But one of my teachers suggested the book by G. H. Hardy, titled Divergent Series. Recently, I was lucky to get hold of this book in our college library. In the preface of this book, J. E. Littlewood quotes Abel:
Divergent Series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.
Also, I came across an article by Christiane Rousseau, titled Divergent series: past, present, future, but the point of view presented there is limited to differential equations and dynamical systems.
As per my knowledge, Riemann's Zeta Function is an important historical example of divergent series. But I don't know as of now whether people doing research in Analytic Number Theory are still interested in general theory of Divergent Series.
I want to know that if there are Number Theorists doing research in Divergent Series. In case there are people doing research in this field, what are the topics of their interest?
Best Answer
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).