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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?

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    $\begingroup$ I think, now this field is called 'Tauberian theory'. $\endgroup$ – Fedor Petrov May 7 '16 at 9:50
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    $\begingroup$ The Riemann zeta-function at $s=1$ diverges, but elsewhere the zeta-function itself is not divergent; its values beyond ${\rm Re}(s) > 1$ come from analytic continuation, not from any theory of divergent series. $\endgroup$ – KConrad May 7 '16 at 12:28
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    $\begingroup$ I am talking about how you should think about the zeta-function today, or for the last 150 years for that matter, not how Euler worked with it. $\endgroup$ – KConrad May 7 '16 at 12:30
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    $\begingroup$ @KCo, can't one interpret analytic continuation as a technique for summing divergent series? $\endgroup$ – Gerry Myerson May 7 '16 at 13:11
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    $\begingroup$ @GerryMyerson ... not in the sense found (e.g.) in Hardy's book. If a series diverges to $+\infty$, then, after applying any regular summation method, it still diverges to $+\infty$. Summation methods are useful when a series diverges by oscillation. $\endgroup$ – Gerald Edgar May 7 '16 at 17:00
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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).

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    $\begingroup$ Just to add one other important case where one now knows analytic continuation: for the $L$-series attached to elliptic curves defined over $\mathbb Q$ (Wiles's theorem). The impact of the work of Wiles et al. on current research is huge. $\endgroup$ – Joe Silverman May 7 '16 at 13:28
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    $\begingroup$ Can you please elaborat, how analytic continuation replaces the classical treatment of divergent series by something more rigorous. $\endgroup$ – rationalbeing May 7 '16 at 17:45
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    $\begingroup$ @rationalbeing: As an example, it allows you to make sense of Ramanujan's "formula" $1 + 2 + 3 +\cdots = -1/12$. Namely that, once one has obtained an analytic continuation of the Riemann zeta function $\zeta$, we have $\zeta(-1) = -1/12$. $\endgroup$ – Daniel Loughran May 7 '16 at 19:19
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    $\begingroup$ @DanielLoughran : Since now we have better tools from complex analysis. In your opinion, would it be useful to go through Hardy's book on Divergent Series? $\endgroup$ – rationalbeing May 7 '16 at 19:54
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    $\begingroup$ @rationalbeing: Unfortunately I have not read Hardy's book, so I cannot advise you there. However, my personal opinion and gut feeling is that things have moved on a lot since Hardy wrote this book, and if you are interested in doing modern mathematical research then divergent series are not really the way to go. I would recommend that you learn some analytic number theory instead. A good introduction for me was "Apostol - Introduction to Analytic Number Theory". More advanced topics at the forefront of modern research can be found in "Iwaniec, Kowalski - Analytic Number Theory". $\endgroup$ – Daniel Loughran May 7 '16 at 21:21
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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.

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  • $\begingroup$ Is this result on theta function discussed in Hardy's book on Divergent Series? If not, then in which book? $\endgroup$ – rationalbeing May 7 '16 at 17:49
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    $\begingroup$ The end of your answer makes no sense: you write about $x$ being irrational but then you give an estimate that has no explicit dependence on $x$ anywhere. What are $q$ and $r$, and how are they related to $x$? Don't assume everyone is going to look up Hardy's book to figure out the notation. $\endgroup$ – KConrad May 7 '16 at 18:21
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    $\begingroup$ Sum of $\mu(n)$ is not $o(1)$, it's $o(x)$... $\endgroup$ – Wojowu May 7 '16 at 19:29
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    $\begingroup$ @johnmangual Can you please make your answer more coherent? In your opinion, is Tauberian Theorems the main take away from Hardy's book on Divergent Series? $\endgroup$ – rationalbeing May 7 '16 at 19:58
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    $\begingroup$ @rationalbeing I can't think of many people who specifically study divergent series or any main take-away of this book. Hardy seem to collect as many result as he can into one place. I think Hardy's book shows coherent discussions can be made about divergent series while studying other topics. $\endgroup$ – john mangual May 7 '16 at 20:05

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