Ramanujan (1887–1920) was only less reclusive than  Turing.  His works, while only briefly and long ago in the mainstream of mathematics, are, even by today’s standards, beacons, neither elusive nor eccentric nor mere sirens. He had an uncanny and often accurate intuition in number theory, where intuition is usually misleading if not outright deceptive. Indeed, he shared much of Fermat’s approach to number theory. Both detested formal proofs and avoided them where allowed by merely asserting deep and correct results. For both, method was more satisfying than proof. To continue the analogy between Ramanujan and Fermat, neither was seduced into philosophical works, unlike Descartes,  Newton,  and Pascal. Ramanujan’s mathematical works pass the usual four-point test of importance: (1) use in other parts of mathematics, (2) use in other disciplines, (3) generation of new research in mathematics and other fields, and (4)universality, if not abstractness.

About five years before his untimely death in India in late April 1920, apparently from tuberculosis, Ramanujan had been studying at Cambridge University with G. H. Hardy, then one of the greatest living mathematicians. In spite of his meager formal education, Ramanujan had great insight into the properties of numbers and the relations between them. For example, he discovered many subtle properties of Bernoulli numbers; highly composite numbers; definite integrals connected in interesting ways with the factorial function; convergent infinite series for accurately expressing various transcendental numbers; useful divergent series; and other deep arithmetical functions. Nevertheless, his usually perceptive intuition concerning prime numbers was somewhat flawed. To his credit is the correctness of one of his complex tau-function conjectures: Put &tgr;* ( n ) = &tgr; ( n ) n ^{- 11&slash;2}. Then &tgr;* ( p ) < 2 for every prime p. (I conjecture that &tgr;* ( n ) > 2 for infinitely many bi-composites n, such as n=799.)

This book is a well-written, detailed, possibly definitive biography of Ramanujan. It includes his turbulent family background, his interactions with numerous other scholars (some scholars, such as G. H. Hardy, receive their own mini-biographies), and his mathematical contributions. This background is embedded in cultures and institutions that have since evolved for the better (some would say they have died). The book goes far beyond the well-known biographical material published by Hardy and others. It even includes material on the mock theta functions, tau function, and highly composite numbers. Also included is more personal material such as Ramanujan’s suicide attempt in London and Hardy’s disposition toward homosexuality. Much material is included on the role of Janaki, Ramanujan’s child bride, during his last year. She may have been India’s first feminist.

The book is divided into eight major sections preceded by a prologue and followed by an epilogue. The sections have such titles as “Ranging with Delight,” “The Search for Patrons,” “Hardy,” “The English Chill,” and “In Somewhat Indifferent Health.” Noteworthy is the mention of applications of Ramanujan’s results. They have applications to statistical mechanics, particle physics, computer science, cryptology, celestial mechanics, pyrometry, crystallography, and the splicing of telephone cables. Particularly valuable are the photographs. They include Ramanujan’s passport photo and photographs of his wife and mother.

I recommend this thorough biography highly, especially as a cultural and historical supplement for graduate students of computer science. I could find few errors. On page 232, after the display of the unique factorization theorem, “composite” should be “highly composite” in two instances; on page 285, “Poussin” should be “Vallee Poussin”; and on page 344 (line 3) the function symbol should be &tgr;. In addition, I found only two enigmatic points. The author never does say whether Janaki, Ramanujan’s wife, is still living; she may well be alive, or have died only recently. Finally, nothing is even hinted at (negatively or positively) in the Ramanujan-Hardy collaboration about Hardy’s homosexual disposition.