Srinivasa Ramanujan

As one reads Srinivasa Rao’s book, the main message that emerges is that the author has an enormous enthusiasm for the work of Ramanujan. Through providing a biography of Ramanujan, presenting some of his primary discoveries, offering biographies of many who were important in the life of Ramanujan, and describing the influences of Ramanujan in the 100 years after his death, the author wishes to motivate others to read more about Ramanujan’s life and to study his work. Thus, perhaps the raison d’etre for writing this book is to provide not only information but also inspiration.

 

The author’s reverence for Ramanujan is captured in many, both short and long, quotations by several mathematicians, including George Andrews, Richard Askey, S. Chandrasekhar, Freeman Dyson, G. H. Hardy, E. H. Neville, Atle Selberg, and the reviewer. Many of these assessments are so thoughtful and eloquent that they have become quite famous. We give an example from a lecture that Dyson gave at the Ramanujan centenary celebration held at the University of Illinois on June 1–5, 1987.

 

 

Robert Kanigel’s beautiful and extensive biography of Ramanujan, The Man Who Knew Infinity, is also frequently referenced. Srinivasa Rao provides an interesting and lucid biography of Ramanujan in the 38 pages comprising Chapter 1.  Several anecdotes and short biographies, in particular, those of Hardy and Chandrasekhar, also add to the enjoyment of the book.

 

Srinivasa Rao’s treatise is partly autobiographical, with many interesting personal reflections. He writes about meetings he has helped to organize, lectures he has given, and CD ROMS he has produced. The author occasionally relates personal complaints and opinions, especially about Indian politicians. Over the years, more than any other person in India (with perhaps one exception), Srinivasa Rao has done more to educate Indians about Ramanujan and his mathematics, and to use Ramanujan as a springboard to stimulate and reform India’s educational policies.

 

The book contains two groups of photographs. The first group has about 50 photos on 19 pages, and the other group offers about 40 photos over 15 pages. Generally, there are 2–4 photographs per page. On almost all pages, the photographs are confined to a small space at the top of the page. The pictures of written documents are either difficult or even impossible to read. It is regrettable that the publisher chose to place these informative and fascinating pictures in such cramped quarters. An attractive feature of the book could have been a beautiful gallery beckoning readers to purchase the book, but unfathomably the publisher chose to relegate the pictures to a secondary status.

 

The mathematical passages contain considerable historical material which will be of interest to many readers. Certain classical facts about hypergeometric series, a particular interest of the author, from both Ramanujan’s first and second notebooks are given. Mock theta functions and, more generally, q-series, the overarching topic in Ramanujan’s lost notebook, are discussed, but not in much detail. 

 

Many of the descriptions are not up to date and do not reflect activity in the last decade or two. For example, on page 97 the author quotes three of the problems that Ramanujan submitted to the Journal of the Indian Mathematical Society. Only the published solutions are mentioned. However, like many of the 58 problems that Ramanujan submitted to the Journal of the Indian Mathematical Society, each of these three problems has been discussed at other places in the literature, and they have had important ramifications. As another example, on page 82 it is mentioned that George Andrews and the reviewer ”have to date brought out three out of the planned four volumes” on the lost notebook. The fourth of five volumes was published in 2013. On page 206, C. A. Reddi, who was related by marriage to Ramanujan’s personal physician, is mentioned in the present tense; he died on March 13, 2014. Richard Askey, who was a renowned expert on much of Ramanujan’s work, is referenced several times in the present tense, e.g., on page 165; Askey died on October 9, 2019. Many further examples can be given. With different phrasings, some passages are repeated many times. It therefore seems that several portions of the volume were written over a long period of time and that they were not completely updated or checked for repetition.

 

There are several errors in the book. We give two examples. On pages 13 and 88 the conditions for the validity of the definition of the Riemann zeta function

 

\( \zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}, \mbox{Re} s > 1 \),

 

are, respectively, given as s > 0 and s ≥ 1. Unfortunately, there are further mathematical mistakes. On page 99, the author, in reference to a passage in the fifth volume that the reviewer wrote on Ramanujan’s notebooks, states, "This volume, however, should be regarded as the closing chapter on Ramanujan’s notebooks..." Instead, the quotation should read: "This volume, however, should not be regarded as the closing chapter on Ramanujan’s notebooks..."

 

Except for some of the mathematical passages, most of Srinivasa Rao’s book can be read by a general audience. Even those familiar with Ramanujan’s life and his mathematics will find something new and interesting. For example, the reviewer learned new information about the statues, sculptures, and busts of Ramanujan that have been created. This is an unusual, idiosyncratic book that can be read in spurts. For those desiring to learn more about Ramanujan, this book can serve as an inspiration to do so.

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The reviewer taught at the University of Illinois at Urbana-Champaign for 52 years; he is now Professor Emeritus. He published five books with Springer on Ramanujan’s notebooks (1985, 1989, 1991, 1994, 1998). He and George Andrews published five books with Springer on Ramanujan’s lost notebook (2005, 2009, 2012, 2013, 2018). The reviewer has advised 37 doctoral students, most of whom found proofs for claims that Ramanujan made in his (earlier) notebooks or lost notebook.  berndt@illinois.edu