# Guidelines for learning about Ramanujan's work?

It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested in the works of Ramanujan, I was wondering if it would be of any help if I studied this "Synopsis".

Moreover, I would appreciate it if someone who has experience with learning about Ramanujan's work (such as his "Notebooks" and his "Lost Notebook") could explain to me what is a good way to approach his works. What are the prerequisites? Would I be able to comprehend his works after having taken courses on, say, Analytic Number Theory and (Real/Complex) Analysis? Is it better to read his notebooks (supposedly after having taken these courses) right away, or start of with books such as "Ramanujan: twelve lectures suggested by his life and work" and "Number theory in the spirit of Ramanujan" (written by Hardy and Berndt respectively)? Should I learn about modular forms beforehand? Etcetera.

Thanks,

Max

• My understanding (and I have no idea about how accurate this is) is that "Synopsis" consists of a list of results, with no proofs, and Ramanujan was only helped by this because he was forced to prove them all himself. As a modern student I think this would be crippling, or at least a waste of time, for all but the most amazing prodigies (there is simply too much mathematics these days to reprove it all yourself) and there is no reason to put yourself through this. – Qiaochu Yuan Feb 1 '11 at 21:43
• @ Qiaochu: hm ok, thanks. I somehow felt that it would not be a very good idea to study the "Synopsis" and expected someone to confirm this idea. We are now left with the second part of the question. – Max Muller Feb 1 '11 at 22:17
• Guidelines for learning about Ramanujan's work ? - Forget everything you've ever learned, and embrace absurdity. – Lucian Jul 5 '15 at 12:16

I know one person who read (at least parts of) Carr's Synopsis when they were young; they are now an exceptionally strong mathematician, but I don't think there is a causal relationship. Rather, there is a correlation between their early interest in mathematics and their independence, and their interest in reading unusual original sources like Carr's Synopsis.

Regarding the question of reading original sources (especially old ones): I think this tends to be discouraged, and the reasons are related to Qiaochu's comment --- it is hard to both learn everything that is necessary to do modern mathematics and spend time reading all the original sources. On the other hand there are advantages that can come from reading older sources that are often not mentioned: for example, one can find points of view or topics or techniques that are no longer emphasized, or are even forgotten, which can sometimes be helpful in a very specific way, and other times helpful in a more general inspirational sense.

To speak of myself for a moment, when I was learning mathematics I read many older and somewhat unusual sources, and I think that there are advantages to this, as well as the disadvantages mentioned above. I think that I have an above average interest in the history of mathematics, and of the historical development of mathematical ideas, and so it was enjoyable and profitable for me to do this. But if I had a different mathematical temperament this wouldn't have been a sensible choice; it was a reflection of my own mathematical personality.

If you find it interesting to read older sources, or you think it might be interesting to try, then I would suggest that you just look at them and see what you get out of it. Just bear in mind that you won't be able to learn modern mathematics in this way, so you wouldn't want such reading to be your only reading by any means; rather, you might want to regard it as a supplement to your regular study of modern expositions.

As for learning specifically about Ramanujan's work, the best place to start is Hardy's book on Ramanujan. This is a wonderful book.

On the other hand, to approach Ramanujan's legacy in a modern context, the best thing to do is to learn the theory of modular forms. For this, Serre's Course in arithmetic is a very good source. (Serre's book has three parts which are only loosely connected. So you can read the section on modular forms without having read the other sections carefully beforehand.)

• @Matt E: Hardy's book on Ramanujan: What is the proper name of the book. – anonymous Feb 1 '11 at 23:41
• @Chandru1: Dear Chandru, It's the book mentioned in the question. Regards, – Matt E Feb 2 '11 at 1:00
• @ Matt E: Thank you for this answer. I heared Course in arithmetic is pretty terse. If this is true, I think I would like to read a more readable treatment on modular forms. What do you think about A Classical Introduction to Modern Number Theory, by Ireland and Rosen? The index says it contains material on modular forms, albeit only a few pages. And what about A first Course in Modular Forms by Diamond or Modular Functions and Dirichlet Series in Number Theory by Apostol? – Max Muller Feb 2 '11 at 20:22
• @Max: Dear Max, I think Ireland and Rosen is a wonderful book, but (unless my memory is very faulty) I don't think you will learn modular forms from it. It's true that Serre's book has a certain terseness to it, but I think that Serre's chapter on modular forms is the least terse of the three chapters. Another thing I like about it is that it restricts to what is called "level one", which means that key ideas such as Hecke operators can be introduced, without all the attendant technicalities that arise if you allow arbitrary level. (These technicalities are important, but they're not ... – Matt E Feb 3 '11 at 3:37
• ... what you should focus on first.) So I think that for modular forms, you should at least try Serre first, before looking for other texts. (I've looked at Apostol long ago, but I have only the dimmest recollection of it, so I can't say much.) Actually, there is one other book that I would recommend, one which may be quite obscure now, but should be in any reasonable university library: Ogg's book Modular forms and Dirichlet series, at least the first half. This book explains the relationship between functional equations (e.g. of the $\zeta$-function) and the transformation properties ... – Matt E Feb 3 '11 at 3:45

I have my own take on approaching Ramanujan's work. My advice is to look at the two volumes of his Noteboks and the Lost Notebook. You may find these in some university libraries, or perhaps online. Read them selectively several times. I found his identities involving his theta functions, Lambert series and modular equations to be most interesting. You may find other topics that interest you. You can supplement this with Berndt's five volumes of editing Ramanujan's Notebook and the four volumes of Andrews and Berndt editing the Lost Notebook. The only prerequisites are power series, a bit of calculus, and college algebra. Good luck.

• +1 and you may also include "Collected Papers of Ramanujan". It has some of most clever proofs from Ramanujan. – Paramanand Singh Nov 28 '18 at 9:08

In addition to Hardy's book on Ramanujan, I would also recommend Ramanujan's collected works, edited by Hardy, P. V. Seshu Aiyar, and B. M. Wilson. Both of these are now published by the AMS - look on ams.org and search for "Ramanujan" in their books. You will uncover a large number of (to me) fascinating publications. I prize my original Chelsea editions of these two works.

The current name to look for in books about Ramanujan is Bruce C. Berndt. I find it amazing the amount and quality of the work he has done.

• To elaborate: Berndt has a series of books published by Springer-Verlag, which discuss the contents of Ramanujan's notebooks, as well as ways to establish them in the spirit of Ramanujan. He also has a number of papers that elaborate on Ramanujan's results. – J. M. is a poor mathematician Nov 26 '11 at 6:31

It is better to go through elementry number theory first .then one can start with hardys book. One can understand Ramanujans notebook if he first goes through elementary ones.

The "Synopsis of Pure and Applied Mathematics" is precisely what it claims to be. It is an abbreviated form of other nineteenth century texts for which it provides reference lists. You are not going to understand the contents of this synopsis properly unless you read these other texts as well, all of which, being 150+years old, are freely available online these days. However, this synopsis is not going to help very much in studying Ramanujan's own work. Instead, it will provide you with a better understanding of undergraduate mathematics at Cambridge in the nineteenth century. If you want to study Ramanujan's work, use Berndt's volumes which detail the notebooks. Berndt lists references that you should dip into to prove formulae that he doesn't prove himself. Before you start, learn some complex analysis and read up on modular forms from Apostol or Serre. Then keep Hardy and Wright at close hand. Do not look at Hardy's volume "Ramanujan", as this is too advanced for the novice and omits the full detail found in Berndt's books. You'll need to refer to Lorentzen and Waaderland's book on the convergence of continued fractions also. After studying the notebooks, you will be able to look at the collected papers (also edited by Berndt) which also have ample reference lists including important books like Andrews' Theory of Partitions which contains Rademacher's improvement of the Hardy-Ramanujan asymptotic partition formula which Selberg also discovered as a teenager.