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Collected Papers of Srinivasa Ramanujan

G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, editors
Publisher: 
Cambridge University Press
Publication Date: 
2016
Number of Pages: 
355
Format: 
Paperback
Price: 
49.99
ISBN: 
9781107536517
Category: 
Textbook
[Reviewed by
Michael Berg
, on
03/23/2016
]

There’s even a movie about him now: The Man Who Knew Infinity (here is the trailer). The title is from the well-known book by Kanigel, and there is even a (serious) documentary available. George Andrews and Béla Bollobás, among others, are prominently featured in the latter, so we can safely trust that it is safe viewing for mathematicians. Indeed, all kidding aside, it is quite a good documentary and conveys the mystery and magic of this most unusual of titans of number theory.

Everyone knows the facts, if only rudimentarily: essentially self-taught, the young Ramanujan sent a letter to G. H. Hardy at Cambridge containing a number of stunning mathematical assertions and formulas that the author hoped would get him airplay in the upper circles of serious mathematical scholarship in England. Hardy describes his reaction in inimitable fashion, the thrust of the tale being as follows. At first he decided the letter was just another crank communication, put it aside, and went about his business of reading examination papers while watching a cricket match on the Cambridge cricket grounds. But he was haunted by what he had read: some of the formulas were correct and known, but highly non-trivial; others were not known to him but had the appearance of being right; and then there were others that looked altogether novel and … well, what was more likely, Hardy argued, a fraud of genius or a genuine untutored genius coming from “the great subcontinent”?

He took the matter up with Littlewood and indeed they found that they could prove some of Ramanujan’s new claims, and others were eminently plausible: with that added to the presence of some deep results that were already known but not to Ramanujan, of course, they decided that there could be no doubt — Ramanujan must be the real article. In due course he was brought to Cambridge from India, was elected a Fellow of the Royal Society, and for a while worked with Hardy and Littlewood, with his unparalleled results properly publicized. He was (relatively soon) taken ill, returned to India and presently died under the care of his wife. In the last years of his life, during his sickness, he composed what is now called his “Lost Notebook,” in which he expounds, among other things, his theory of mock theta functions. Hardy ‘s obituary of Ramanujan is one of the most poignant things Hardy ever wrote, perhaps in part because it also reveals so much about the ultimately tragic figure of Hardy himself (for example his all but pathological war on God).

The book under review, edited by, among others, Hardy, is a re-issue by Cambridge University Press of the collected papers of Ramanujan. It is introduced by a pair of “Notes” which are sources of wonderful information about Ramanujan in their own right, both as regards his life and his mathematics. After that it is all about his mathematics: thirty seven articles on number theory, infinite series, integrals, and combinatorics. It is all stunning, both by virtue of the content of these articles and because of the idiosyncrasy of their author. After all, Hardy claimed that his attempts to teach Ramanujan even complex analysis were only marginally successful, since the latter’s own methods were as powerful as they were heterodox: he didn’t need much of what modern mathematics as preached by Hardy had to offer. The book ends with a pair of appendices, respectively, “Notes on the papers” and “Further extracts from Ramanujan’s letters to G. H. Hardy.” All fabulous stuff.


Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preface
Notice P. V. Seshu and R. Bamachaundra Rao
Notice G. H. Hardy
Part I. Papers:
1. Some properties of Bernoulli's numbers
2. On question 330 of Prof. Sanjana
3. Note on a set of simultaneous equations
4. Irregular numbers
5. Squaring the circle
6. Modular equations and approximations to π
7. On the integral [...]
8. On the number of divisors of a number
9. On the sum of the square roots of the first n natural numbers
10. On the product [...]
11. Some definite integrals
12. Some definite integrals connected with Gauss's sums
13. Summation of a certain series
14. New expression for Riemann's functions [...]
15. Highly composite numbers
16. On certain infinite series
17. Some formulae in the analytic theory of numbers
18. On certain arithmetical functions
19. A series of Euler's constant y
20. On the expression of a number in the form of ax2+by2+cz2+du2
21. On certain trigonometrical sums and their applications in the theory of numbers
22. Some definite integrals
23. Some definite integrals
24. A proof of Bertrand's postulate
25. Some properties of p (n), the number of partitions of n
26. Proof of certain identities in combinatory analysis
27. A class of definite integrals
28. Congruence properties of partitions
29. Algebraic relations between certain infinite products
30. Congruence properties of partitions
29. Algebraic relations between certain infinite products
30. Congruence properties of partitions
Part II. Papers Written in Collaboration with G. H. Hardy:
31. Une formule asymptotique pour le nombre des partitions de n
32. Proof that almost all numbers n are composed of about log log n prime factors
33. Asymptotic formulae in combinatory analysis
34. Asymptotic formulae for the distribution of integers of various types
35. The normal number of prime factors of a number n
36. Asymptotic formulae in combinatory analysis
37. On the coefficients in the expansions of certain modular functions
Questions and solutions
Appendix 1. Notes on the papers
Appendix 2. Further extracts from Ramanujan's letters to G. H. Hardy.

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