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The Mathematical Legacy of Srinivasa Ramanujan

M. Ram Murty and V. Kumar Murty
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

It is impossible to be a number theorist and not to be a huge admirer and fan of Srinivasa Ramanujan. And even if you’re a mathematician espousing another allegiance (even if you’re a logician!) it is still extremely unlikely — No! It’s impossible! — that you’re not at least cognizant of the myth as well as the magic of this wonderful mathematician whose short life ended so tragically in 1920, when he was only 32.

Much of what we know about Ramanujan is owed to his mentor, advisor, collaborator, and friend G. H. Hardy. By way of an illustration, Hardy presented Ramanujan to a Harvard audience, some sixteen years after the latter’s death, with the piquant phrase, “I have to help you to form some sort of reasoned estimate of the most romantic figure in the recent history of mathematics ...” After this Hardy recalled the famous episode of his and Littlewood’s realization that they had before them one of the very rarest of jewels: “I can still remember with satisfaction that I could recognize at once what a treasure I had found.”

We know the details, of course: the large envelope from India, sent to Hardy at Cambridge, at first tossed aside, then beckoning. Hardy scanning the contents: the list of formulas Ramanujan, still a complete unknown, without a completed education in mathematics, had sent him, and Hardy asking the question: which was more likely, actual genius or a hoax of genius? Indeed the formulas in question were either correct and known, off-beam but indicating brilliance, or new and extremely tantalizing, suggesting anything but a hoax. The tale continues, with Hardy presently collecting Littlewood and the two of them going at proving some of Ramanujan’s new formulas, many of the highest order of difficulty. Finally Hardy agreeing that, indeed, he had been approached by one whom he would later describe with the words, “I have never met his equal, and I can compare him only with Euler or Jacobi.”

The story continues, of course. Ramanujan eventually came to England and studied with Hardy, who graciously observed that while Ramanujan was unschooled in many things every modern mathematics student in the West would know, e.g., complex analysis, it was often Ramanujan who taught Hardy deep new mathematics, much of which then needed to be restructured so as to become rigorous, complete, and communicable. Naturally, Ramanujan blossomed mathematically, becoming a Fellow of the Royal Society as well as earning a Cambridge research degree with Hardy and Littlewood as advisors. His health was not good, however, and got dramatically worse in England, so that in due course he returned to India, working on mathematics to the day he died.

This is the bare-bones outline of Ramanujan’s life, and the interested reader simply cannot do any better than to read the biography by Robert Kanigel, The Man Who Knew Infinity, and of course anything and everything Hardy wrote about Ramanujan. Or, for those who prefer long wave-length radio waves, there is the March 22, 1988 “Nova” documentary, “The Man Who Loved Numbers.” All excellent.

But this is just part of the story: what about Ramanujan’s work and its impact on mathematics? To address this, what the authors of the book under review have done is to present, in their ten-page first chapter, a wonderful essay with both fine biographical aspects and evocative indications of how Ramanujan’s work fits into the bigger picture of contemporary number theory and algebraic (and particularly arithmetic) geometry. They touch upon such things as the famous circle method, the normal order method, and Fourier coefficients of modular forms (in connection with which there arise the three Ramanujan conjectures and the famous “tau” function). Thereafter they address the interface with algebraic geometry engendered by the work of, particularly, André Weil and Pierre Deligne and the appearance of, e.g., the Riemann Hypothesis for zeta functions of varieties over a finite field. The latter is of course fitted into the framework of the Weil Conjectures, and it was Deligne who famously settled them in the 1970s.

The introductory chapter to the book then turns to a somewhat more discursive discussion of certain elements of Ramanujan’s life, including the 1729 affair. (See p.6, if this isn’t known to you, or else just google the number!) Finally there is a brief discussion of Ramanujan’s particular role as an Indian scientist, including various interesting comments by the astrophysicist Subramanian Chandrasekhar.

After this, the mathematics proper begins. The Murtys’ goal, after all, is to present Ramanujan’s mathematical legacy to a broad audience, and the thrust of the book is a set of eleven chapters discussing exactly that. It’s stunning stuff, of course, and entirely irresistible to a number theorist: it starts with “tau” and the conjectures (including elliptic curves, l-adic representations, and modular forms with a vengeance), then the circle method, transcendence (including Rogers-Ramanujan), the partition function, mock theta functions, highly composite numbers, probabilistic number theory (where we encounter the normal order method) and finally Sato-Tate. It’s overwhelming, really, and overwhelmingly beautiful.

This wonderful book, published in celebration of the 125th anniversary of Ramanujan’s birth, can’t help but persuade many an aspiring student to opt for number theory (modulo the kid possessing the right disposition, including no small amount of patience), just as it can’t help but inspire more seasoned veteran’s in the field. As far as the aforementioned mathematicians expressing another allegiance are concerned, the book will utterly charm you, given its accessibility, style, structure, and depth. It’s a great pleasure to read, and it’s fine scholarship.



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