Divergent Series as a topic of research 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?
 A: 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).
A: There are different procedures for extracting a real number from a divergent series.  We tend to say:
$$ 1 - 1 + 1 - 1 + 1 - \dots = \frac{1}{2} $$
and we use the Cesaro averages as a rule to .  Lately I am personally of the school that the partial sums are of two values;
$$ \sum_{n = 0}^N (-1)^n = \left\{  \begin{array}{cl} 1 &  n \text{ is odd} \\ 0 & n \text{ is even} \end{array} \right. $$
Then there is some type of limit, but the result is not a real number - an element of $\mathbb{R}$ - it is a distribution.  We could try to say:
$$ \lim_{N \to \infty} \sum_{n = 0}^N (-1)^n \approx \frac{1}{2}\delta_{0} + \frac{1}{2}\delta_1$$
Divergent series come up as part of regluarization in physics, which are attempts to get finite numbers when the aren't any.  As an avid reader of hep-th I can attest to physicsts at prominent institutions, casually assigning finite value to bad divergent sums and integrals.
The methods in Hardy's divergent series book or the theory of distributions will formalize such an intuition.

Many discussions end the conversation when they establish a particular series is divergent.  Hardy's approach seem to study how the series diverges and establish a rate of divergence.  Here is one of his result on the theta function. Let $x \notin \mathbb{Q}$, and $q = re^{2\pi i n x} \in \mathbb{D}$ then 
$$ 1 + 2 \sum q^{n^2} = o\left( \sqrt[4]{\frac{1}{1-r}}\right) $$
Although I am slightly confused since if the continued fraction digits are bounded  - $x = [a_0; a_1, a_2, \dots]$ with $|a_n| \leq M$ he shows:
$$ \left| 1 + 2 \sum q^{n^2} \right| \asymp 
 \sqrt[4]{\frac{1}{1-r}} $$
This is from a the collected papers of Hardy, [1]

This derivation is from page 4 of Hardy's divergent series:
\begin{eqnarray*} \sum (-1)^{n-1}\frac{1 - \cos n \theta}{n^2} &=& \sum \frac{(-1)^{n-1}}{n^2} \sum (-1)^k \frac{(n\theta)^{2k+2}}{(2k+2)!} \\
&=& \sum (-1)^k \frac{\theta^{2k+2}}{(2k+2)!} \sum \frac{(-1)^{n-1}}{n^{2k}} \\
&=& \frac{1}{2}\theta^2( 1 - 1 + 1 - 1 + \dots) \\
&=& \frac{\theta^2}{4} \end{eqnarray*}
Then Hardy begins to express concern this derivation can't hold in general:
$$ \sum (-1)^{n-1} \frac{f(\theta)}{n^2} = \frac{\pi^2}{2} a_0 + \frac{\theta^2}{2} a_1 $$
This is not true for $f(\theta) = a_0 + a_1 \theta^2 + a_2 \theta^4 + \dots$ so what gives?  
Even though $\sum (-1)^n = \frac{1}{2}$ is clearly right in many circumstances, Hardy shows it can lead to false statements.  How do we reject something which is so right and intuitive?

Lastly, Hardy shows the prime number theorem.  The odds of an n-digit number being prime is $\frac{1}{\text{# of digits of } n}$.  Or the average of the Mobius function is $0$... among square-free number half of them have an even number of factors and half of them have an odd number of factors:
$$ \frac{1}{N} \sum_{n \leq N} \mu(n) = o(1) $$
These are proven from tauberian theorems although there are several other steps involved.
