Lasting Legacy of Ramanujan: Transcending Kanigel’s Canvas

Robert Kanigel, the author of *The Man Who Knew Infinity* *The Hindu*

While visiting India, in an interview given to The Hindu published on 26 December 2011, Robert Kanigel described his book The Man Who Knew Infinity on Ramanujan as a `narrative non-fiction’. In his words, “In the case of Ramanujan, what I wrote was the first western biography…. In some respects, I consider this almost a dual biography about Ramanujan and Hardy…. For me, Hardy played such an important role that their chemistry, their tension, their friendship, their relationship played a central role mathematically and personally in Ramanujan’s life. And I felt it was really important for the reader to come to understand Hardy as well as Ramanujan.”

Biographies of mathematicians, barring a few exceptions, are hardly known to be popular, but Kanigel has trodden that seemingly impossible path some three decades earlier to produce the most elegantly awe-inspiring tale of the last century. His book, a psycho-analytic dual biography of Ramanujan and Hardy, in its odyssey of capturing the quintessential Ramanujan through his romantic sojourn, has won many accolades since its publication, and made the author a distinguished international celebrity. In the USA, Robert Kanigel’s book was first published in 1991. It was highly praised and was translated into German in 1993, Japanese in 1994, Korean in 2000, Chinese in 2002 and again in 2008, Italian in 2003, Thai in 2007, Greek in 2008 and Polish in 2016. In India, an English-language edition of this book (by Rupa Publications) was first published in 1992. For a long time, no full-length life-story of Srinivasa Ramanujan was available in most of the Indian languages—a biography that is comparable to Kanigel’s magnum opus, that would do justice to the phenomenal yet tragically short life of this self-taught genius of inexplicable originality, and would simultaneously lay before the reader an informed sketch of his stupendously huge mathematical oeuvre that he left behind as a legacy for the generations to come, and its ever-growing importance in mathematical fields.

M.S. Raghunathan TIFR Alumni Association

Meanwhile, during a program held at the Madras University Centenary Auditorium, Chennai, on 26 December 2011, inaugurating the year-long celebrations of the 125th birth anniversary of Ramanujan, the then Prime Minister of India, Dr. Manmohan Singh declared December 22, Ramanujan’s birthday, as the National Mathematics Day and 2012 as the National Mathematics Year. Releasing a commemorative stamp on Srinivasa Ramanujan to mark the occasion, Dr. Singh referred to mathematics as “the mother science,” and pointed out that “the Ramanujan story illustrates the inadequacy of the university evaluation system in the early decades, while at the same time it shows that the system displayed enough flexibility to take care of mavericks like him. A genius like Ramanujan would shine bright even in the most adverse of circumstances, but we should be geared to encourage and nurture good talent which may not be of the same calibre as Ramanujan’’. A hugely improved new edition of the Ramanujan Notebooks, made with the help from the archiving and digitising team at the Roja Muthiah Research Library (RMRL) in Chennai, was released by TIFR (Tata Institute of Fundamental Research) on the occasion. In this program, Robert Kanigel was honoured by Tamil Nadu Governor K. Rosaiah for his excellent biography of Ramanujan. An Organising Committee with M.S. Raghunathan, the then President of the Ramanujan Mathematical Society (RMS) as chair, and Dinesh Singh as secretary, was formed to formulate and implement programs and projects as part of the observance of the National Mathematics Year. A National Committee with the then Minister for Human Resource Development, Mr. Kapil Sibal, as the chair was formed to supervise the activities of the organising committee. Vice-chair of the national committee M.S. Raghunathan outlined the programs planned for the next one year as part of the celebrations. Various mathematical activities were planned to cater to the diverse sections of society all over the country. One program meant for the public was a lecture tour by Kanigel covering Mumbai, Chennai, Bangalore, Hyderabad and Delhi, which took place eventually in due course of time.

Dr. Manmohan Singh and Kapil Sibal at the inauguration of Ramanujan’s 125th birth anniversary celebrations *The Hindu*

At about the same time another development took place. From the year 2012, International Centre for Theoretical Sciences (ICTS) of TIFR had started a yearly Lecture Series named after Ramanujan. Peter Sarnak of Princeton University was the first lecturer who spoke on “The Generalized Ramanujan Conjectures and Applications” on 21 May 2012 at TIFR Mumbai.

During December 17–22, 2012 an International Conference on mathematics related to and influenced by Ramanujan’s work was organised by University of Delhi. An International Scientific Committee chaired by Bruce Berndt suggested a list of eminent Ramanujan Scholars to be invited as speakers. This grand conference, named `The Legacy of Srinivasa Ramanujan’, was attended by a large number of mathematicians and Ramanujan enthusiasts from all over the world.

Meanwhile, the organising committee to commemorate the 125th birth anniversary also took up the project of getting Kanigel’s book translated into ten Indian languages under the auspices of RMS. The first one to come out was in Bengali. It was published jointly by RMS and the National Book Trust, India, and was formally released on 19 July 2013 at Kolkata, by Amartya Sen, the Nobel Laureate economist, who was once the Master as well as a Fellow of Trinity, the College of Cambridge, where Ramanujan once collaborated with Hardy. Within a month, the Malayalam translation was released by the Chief Minister of Kerala. The Kannada and the Tamil translations were published on 22 December 2013, the birthday of Ramanujan now declared as the National Mathematics Day in India. Shortly thereafter, the Odiya, Marathi and Gujarati translations were also published. Three other translations which were initiated are yet to be published.

Amartya Sen releasing the Bengali translation of *The Man Who Knew Infinity* (L to R: Bimal Roy, Parthasarathi Mukhopadhyay, Amartya Sen, M.S. Raghunathan, Bratin Dey) Parthasarathi Mukhopadhyay

However, water has been flowing relentlessly through the Kaveri since the book by Kanigel was first published in 1991. And with the passage of time, the information related to Ramanujan, as one may find in Kanigel’s well-researched book, comprehensive and up-to-date as it was during its time of publication, appeared to require further updating here and there. This is why, while working on the Bengali translation, it became almost a moral compulsion to try one’s level best and shed some light on the recent boom in the mathematical arena influenced by Ramanujan’s mathematics, as well as some other socio-academic developments related to his name and life story that have taken place since then. It is with this mission at the back of mind, an addendum was proposed to this biography of epic proportions, which was duly accepted and appreciated by Robert Kanigel and finally got appended to the Bengali translation. Later, in December 2013, an abridged modification of the same was published in the Mathematics Newsletter of the Ramanujan Mathematical Society. The present one, a bouquet of various information, is a slightly revised and updated version of that article. It is divided into two parts. The first part updates some of the available information on the various academic and socio-cultural facets relating to the growing impact of Ramanujan’s life in our society at large, while the second one is a brief non-technical survey on the influence of Ramanujan’s works on present-day mathematical research.

Part I

At the outset, let us focus on the not-so-mathematical, and rather socio-cultural aspects related to Ramanujan that took place during the last three decades or so. In an article entitled “The Meaning of Ramanujan Now and for the Future” commemorating the 123rd anniversary of Ramanujan’s birth on December 22, 2010, George E. Andrews, while paying homage to “this towering figure whose mathematical discoveries so affected mathematics throughout the twentieth century and into the twenty-first,” pointed out to the reader that “whenever we remember Ramanujan, three things come most vividly to mind: (1) Ramanujan was a truly great mathematician; (2) Ramanujan’s life story is inspiring; and (3) Ramanujan’s life and work give credible support to our belief in the Universality of truth.” In this article, Andrews aptly points out that “almost anyone interested enough in Ramanujan to be reading these words knows the broad outline of Ramanujan’s life.” There have been many biographies of Ramanujan written before and after the Kanigel book, in English and also in other regional Indian languages. One such notable book in the recent past was by K. Srinivasa Rao (1998). Three short biographies of Ramanujan by S. Ram (2000) in English, by Satyabachi Sar (2000) and Ranjan Bandyopadhyay (2009) in Bengali are also worth mentioning. Another one, entitled The Mathematical Legacy of Srinivasa Ramanujan was published in October 2012. Written by eminent Ramanujan scholars and mathematicians, M. Ram Murty and V. Kumar Murty, this book has several chapters devoted to Ramanujan’s conjecture and its impact on 20th- and 21st-century mathematics. In any such book on Ramanujan, mention of Kanigel’s book in the bibliography is a common feature. On the other hand, Kanigel’s passionate research is evident from the similar mentions of almost all the major Ramanujan-related books published in English before. However, there is a notable exception both ways, which seems worth mentioning.

Almost at the same time as Kanigel’s book, Wazir Hasan Abdi, a distinguished professor of mathematics and a Fellow of the Indian National Science Academy, wrote a beautiful book on Ramanujan’s life and work entitled Toils and Triumphs of Srinivasa Ramanujan, the Man and the Mathematician. Apart from containing almost all the information on Ramanujan’s life that one may get in Kanigel’s work, it includes all the contributions that Ramanujan made to the Journal of Indian Mathematical Society before his voyage to Cambridge. Furthermore, in the last section, there is a collection of six survey articles written by experts on various areas of Ramanujan’s mathematical work.

V. Kumar Murty **courtesy** IAS Princeton

Subsequently, Bruce C. Berndt and Robert A. Rankin have published two wonderful books. The first one, called Ramanujan – Letters and Commentary, published in 1995 by AMS (Indian edition by Affiliated East West Press Private Limited, 1997) collects various letters written to, from, and about Ramanujan, and makes detailed commentaries on the letters. The second book, called Ramanujan – Essays and Surveys, published in 2001 by AMS-LMS (Indian edition by Hindustan Book Agency, 2003) is a collection of excellent articles by various experts on Ramanujan’s life and work. The book is divided into eight parts and contains articles about certain individuals who played a major role in Ramanujan’s life. A noted Ramanujan scholar from the University of Florida, Krishnaswami Alladi, says “both books will appeal not only to mathematicians, but to students and laypersons as well”. While writing a review of the Indian edition of the second book in Volume 87 of Current Science in 2004, M.S. Raghunathan remarked that “our curiosity about the great is by no means confined to their lofty pursuits and achievements (perhaps because it is their commonplace experiences that will reassure us of our kinship with them).

R.A. Rankin MacTutor History of Mathematics Archive, University of St. Andrews

And in the case of Srinivasa Ramanujan, that curiosity is greater as the bare outlines of his life have so much of the romantic element. The excellent biography of Ramanujan by Robert Kanigel… is indeed a comprehensive work, yet there remain many unanswered questions. The volume under review provides some facts of human interest not to be found in Kanigel’s work and at the same time caters also to those who seek an introduction to the workshop of the genius.” Highlighting such issues he further noted, “A note by Berndt tells us that the original notebooks are to be found in the Librarian’s office of the University of Madras…. We owe the publications of The Notebooks to the initiative of K. Chandrasekharan, one of our leading mathematicians—a fact that unfortunately finds no mention in much of the extensive writings about Ramanujan.” Part two of the book contains information regarding Ramanujan’s illness and treatment. An article by R.A. Rankin reproduced in the book from the Proceedings of the Indian Academy of Sciences and another by a physician, D.A.B. Young, who did some investigations on this subject for the Royal Society, have considerable information not to be found in Kanigel’s book. Raghunathan pointed out that, “among other things, it appears that the contemporary diagnosis of Ramanujan’s illness was not satisfactory”. Part three of this book has a short biography of Janaki, Ramanujan’s wife, who eventually passed away on April 13, 1994, at the age of 94. An interview given by her to Pritish Nandy has been reproduced in the book, which reveals some glimpses of her personality.

A more recent addition to this list of books is the one from Springer in 2016 titled My Search for Ramanujan: How I Learned to Count. Jointly written by an eminent Ramanujan scholar, Ken Ono, along with Amir D. Aczel (of Finding Zero fame), their “search for Ramanujan ranges over three continents and crosses paths with mathematicians whose lives span the globe and the entire twentieth century and beyond.”

Robert Kanigel, in an interview given to The Hindu in December 2011, observed, “after the book (The Man Who Knew Infinity) came out, there was new theorising about what Ramanujan actually died from. That would have been interesting to bring to a book on Ramanujan written after that period’’. In an article in the Asia Pacific Mathematics Newsletter (April 2012, vol 2, no. 2), K. Srinivasa Rao of The Institute of Mathematical Sciences, Chennai, also pointed out the same. “Mainly due to the efforts of R.A. Rankin, a renowned mathematician, and Dr. D.A.B. Young, a medical doctor, it is now common knowledge, amongst the admirers of Ramanujan, that the cause of the death of Ramanujan was not the then dreaded TB, but hepatic amoebiasis, which was the cause of his illness twice in his younger days, in India. Since TB was diagnosed by (some) doctors in England and in India after his return, as a celebrity, he got the best medical attention and the full-fledged backing of the University of Madras. Since the treatment was done (not for hepatic amoebiasis but) for TB, it led to his premature death.”

In his expertly penned article full of medical jargon written for the Royal Society, Dr. Young pointed out that “unfortunately, no official medical records of Ramanujan’s illness during his time in England have survived, so any attempt at a retrospective diagnosis must depend on information in letters and reminiscences.” And on the basis of such information, he retraced the whole medical history of Ramanujan’s life and arrived at his claim of misdiagnosis of tuberculosis:

Letters that Ramanujan wrote from the Matlock Sanatorium to Ramalingam and to Hardy courtesy The Master and Fellows of Trinity College, Cambridge

Many may wonder that any mystery attends Ramanujan’s illness, for until 1984 it was generally believed that tuberculosis was the cause of death, and that his illness had been explicitly treated as such in the various English sanatoria and nursing homes. However that diagnosis originated, not in England, but in India with Dr. P. Chandrasekhar of the Madras Medical College, who attended [to] Ramanujan from September 1919 until his death on 26 April 1920. The ever-worsening emaciation and pulmonary symptoms that followed his relapse on arriving in India have been powerfully persuasive. The verdict of the English doctors was quite the opposite, but its publication had to wait until Rankin brought the hitherto unpublished material together in his 1984 papers. It quite clearly contradicts, for example, the statement in Ranganathan’s book that `by the end of 1918, it was definitely known that tuberculosis had set in’. It was disappointing, therefore, that Kanigel in his biography, although acknowledging the doubts, nevertheless chose to perpetuate the undue emphasis on tuberculosis.

In his ten-page long analysis of the diverse possibilities that the symptoms suffered by Ramanujan might indicate, Young carefully took into account and examined all the available facts about Ramanujan’s illness and the contradicting opinions of the doctors who attended to him and finally opined that,

Although we have but little information about Ramanujan’s illness, there is much that can be inferred from it. The illness began with an acute episode that was diagnosed as gastric ulcer. Later, the condition eased and the symptomatology must have changed significantly, for this diagnosis was rejected and that of tuberculosis favoured. The sign prompting this was probably the onset of `intermittent pyrexia’, which eventually became the regular nighttime fevers described by Hardy. `Intermittent pyrexia’ is found in a group of otherwise diverse diseases, by far the most important of which at that time was tuberculosis. Expert opinion was accordingly sought from Dr. H. Batty Shaw (1867–1936), a London specialist in consumption and chest diseases. His verdict, given probably in August 1917, was that it was not tuberculosis but metastatic liver cancer, derived, he believed, from a malignancy of the scrotum excised some
years before. Time proved him wrong, but it must be very significant that he did not favour tuberculosis. By the summer, the bouts of intermittent fever had become less frequent, and they ceased altogether during Ramanujan’s stay at Fitzroy Square. Here, doubtless, advantage was taken of the wealth of medical expertise and facilities available in London. The consensus of medical opinion mentioned by Hardy in his November letter (to Dewsbury), namely that Ramanujan had been suffering from some obscure source of blood poisoning, entails that, at a minimum, blood counts were carried out. These are procedures that were, even then, a matter of routine.

Communication to Hardy, from Madras University, of Ramanujan’s death **courtesy** The Master and Fellows of Trinity College, Cambridge

Armed by this hypothesis, he then presented `a diagnosis by exclusion’, which according to him was “limited to clinical detail and diagnostic procedures available in 1918, as presented in the 1917 edition of Munro’s textbook”. Referring to the diseases, that might have caused the dominant symptom of `Intermittent pyrexia’, and eliminating them one-by-one through their would-be-evident manifestation in a blood count he reached the crux of his theory:

The fourth disease is hepatic amoebiasis and the fact that it has been arrived at here by a process of elimination should not disguise the high probability that it could have been the cause of intermittent fever in someone of Ramanujan’s background. For, as an `Always suspect hepatic amoebiasis in a patient with obscure pyrexia, symptoms would be pain with tenderness in epigastrium, enlargement of the liver, and weakness’. Progressive emaciation leading to cachexia is characteristic of the disease. If hepatic amoebiasis was suspected, the effect of emetine for 8–10 days on the patient’s symptoms was the easiest way to confirm the diagnosis; and it remains so today, although metronidazole would now be preferred to emetine.

Elaborating on Ramanujan’s background, in this regard, he pointed out that in the second half of 1906, while attending senior school in Madras, Ramanujan contracted a bad bout of dysentery that forced him to return home for three months. “This was probably amoebic dysentery,” wrote Young, “the most common form in India at the time’’. Amoebiasis, unless adequately treated, is a permanent infection, although many patients may go for long periods with no overt signs of the disease. Relapses occur when the host-parasite relationship is disturbed. Ramanujan experienced such a relapse, I believe, in 1909 when he was nursed by his friend R. Radhakrishna Aiyar. Young then brought into account the `hydrocele’ episode and argued that it was “a scrotal `amoeboma’ rather than a hydrocele,… a lesion arising from the relapse, for amoebae can spread into adjacent tissues in the anogenital area” and “Dr. Shaw’s suspicion that the `hydrocele’ operation was the excision of a malignant growth” seemed to corroborate with this fact. He then pointed out that, “in the spring of 1917 Ramanujan became acutely ill, gastric ulcer was diagnosed. This could have been a recurrence of intestinal amoebiasis, this time in the transverse colon, where it can give rise to symptoms closely resembling those of gastric ulcer, but without dysentery. This could have then led to hepatic amoebiasis, with the changed symptomatology.” Young concluded the article with the following remarks:

Two questions will inevitably arise from the suggested diagnosis of hepatic amoebiasis; namely, why the diagnosis was not made at the time and what prospect of a cure could there have been. Hepatic amoebiasis was regarded in 1918 as a tropical disease (`tropical liver abscess’), and this would have had important implications for successful diagnosis, especially in provincial medical centres. Furthermore, the specialists called in were experts in either tuberculosis or gastric medicine. Another major difficulty is that a patient with this disease would not, unless specifically asked, recall as relevant that he had had two episodes of dysentery 11 and 8 years before. Finally, there is the very good reason that, because of the great variability in physical findings, the diagnosis was difficult in 1918 and remains so today: hepatic amoebiasis `presents a severe challenge to the diagnostic skills of the clinician… [and] should be considered in any patient with fever and an abnormal abdominal examination coming from an endemic area’. This same admonition was made by Savill in 1930 but not in earlier editions of his book (e.g. that of 1918). The treatment available in 1917 was essentially the same as that in 1962, namely administration of emetine with aspiration of any detectable abscess. Emetine alone would probably have cured Ramanujan in 1917, and perhaps as late as his departure for India. Since the abscess must have enlarged rapidly thereafter aspiration would have been essential. Pulmonary amoebiasis responds well to emetine, so that unless secondary infection or empyema had occurred, emetine and aspiration might have effected a cure as late as January 1920, when Ramanujan wrote for a last time to Hardy, full of hope, wanting to subscribe to new journals and still producing mathematics of very high quality.

However, Professor (Dr.) D.N. Guha Mazumder, a noted Gastroenterologist, who is a member of the Task Force of Liver Disease, Indian Council of Medical Research, New Delhi and former Head of the Department of Gastroenterology, Institute of Post Graduate Medical Education and Research, Kolkata, holds contrary views. Referring to the `diagnosis by exclusion’ as proposed by Dr. Young as `far-fetched’, he points out the fact that, at the time under consideration, Madras was quite advanced in tackling tropical diseases, and with blood count data suggesting any possibility of a liver abscess, it seems quite unlikely that the eminent doctors treating Ramanujan would have failed to explore that possibility as well. He further suggests that it is highly unlikely that a case of liver abscess, if not treated properly, would take such a prolonged period of time to worsen through gradual deterioration, rather than a faster manifestation of ultimate decline. He strongly feels that the available data are not sufficient to favour `hepatic amoebiasis’ against `tuberculosis’, for which Ramanujan was treated.

Ramanujan wrote for a last time to Hardy, full of hope, wanting to subscribe to new journals

Let us now turn our attention to the academic and cultural happenings that commemorate Ramanujan in the last three decades. In 1985, a few years before Kanigel took up his project, the Ramanujan Mathematical Society (RMS), an organisation with the aim of “promoting mathematics at all levels” was formed at Tiruchirappalli, Tamil Nadu. In due course of time, the RMS blossomed into one of the most prominent among such bodies in India. Today, the publications of Ramanujan Mathematical Society include the Journal of the Ramanujan Mathematical Society which, started in 1986, was initially a bi-annual journal but has now flourished to publishing four issues per year, and Mathematics Newsletter. Launched in the year 1991, the Mathematics Newsletter is a publication catering to the needs of students, research scholars and teachers, and is circulated free of cost among various academic institutions within India. The Lecture Notes Series is a publication consisting of monographs and proceedings of conferences. From 2014, RMS has also added the Collected Works Series, where seminal lifetime contributions of prolific Indian mathematicians are brought together. At least one prominent mathematical journal bearing the name of Ramanujan came out during the last three decades. Given the name, The Ramanujan Journal and published by Springer since 1997, it is devoted to the areas of mathematics influenced by Ramanujan.

From 2008, Department of Science and Technology, Government of India, instituted the Ramanujan Fellowship, presently worth Rs. 1,35,000 per month with a contingency grant of Rs. 7 lakh per annum and Rs. 60,000 per annum as overhead charge, meant for brilliant Indian scientists and engineers below the age of 40 years to take up scientific research positions in India, especially those who want to return to India from abroad and are not holding any position in the Indian institutes or universities. The duration of the fellowship is for five years. The Ramanujan Fellows may work in any of the scientific institutions and universities in India and apart from the fellowship, they would be eligible to apply for core research grant proposals. From 2005, an annual prize for young mathematicians of age less than 45 years, from developing countries who conduct their research in a developing country, has been created in the name of Ramanujan by the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, in cooperation with the International Mathematical Union (IMU) and with support from the Niels Henrik Abel Memorial Fund, Norway. However, the involvement of the Abel Fund ended in 2012 and it was taken up by DST, Govt. of India. Presently it is worth $15,000. The list of recipients from India till date includes Sujatha Ramadorai in 2006, Amalendu Krishna in 2015 and Ritabrata Munshi in 2018. In order to honour the legacy of Ramanujan and to encourage an early locating and nurturing of the exceptional talents worldwide, who may lack traditional institutional support in the fields of mathematics, engineering and science as a whole, through financial grants and mentorship opportunities the Spirit of Ramanujan STEM Talent Initiative was introduced from 2016. It is headed by Ken Ono, presently the Vice President of AMS and Thomas Jefferson Professor of Mathematics at University of Virginia, and funded by the financial support from Templeton World Charity Foundation. Winners of this international talent search receive up to $5,000 to fund their participation in research programs around the world or individual research with an approved sponsor. The scholarship is granted yearly from 2017 and, to date, there are 32 winners of this award from 12 countries around the globe, some of them as young as eleven years at the time of award.

The Shanmugha Arts, Science, Technology & Research Academy (SASTRA), a private university based in Tamil Nadu, has instituted the SASTRA Ramanujan Prize of $ 10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan. This prize is awarded annually since 2005, at an international conference conducted by SASTRA in Kumbakonam, Ramanujan’s hometown, around Ramanujan’s birthday, 22 December. The joint first recipients were Manjul Bhargava of Princeton University and Kannan Soundararajan of the University of Michigan. Akshay Venkatesh of Stanford University, one of the four 2018 Fields Medalists, received the prize in 2008. Research chairs have also been established at the SASTRA centre at Kumbakonam in honour of Ramanujan—two by the Department of Science and Technology, Government of India and one by City Union Bank Ltd., to encourage research in the field of mathematics. SASTRA purchased the home of Ramanujan in 2003, and has since maintained it as a museum.

V.S. Varadarajan and his wife, Veda **courtesy** UCLA

Another single-room Ramanujan Museum and Math Education Centre was created in 1993 in the premises of the Avvai Academy in Royapuram, Chennai by P.K. Srinivasan, a dedicated school teacher in mathematics, with the help of a benevolent businessman Mr. A.T.B. Bose. In this little-known museum, there’s a treasure trove of pictures, letters and documents which include Hardy’s replies to Ramanujan’s first two letters sent to him in early 1913. In 1999, Srinivasa Rao and R. Jagannathan helped establish at the Periyar Science and Technology Centre, Kotturpuram, a Ramanujan Gallery, which includes a poster-rich `Pie Pavilion’, a multimedia presentation on the life of Ramanujan and a replica of the Ramanujan Museum. A Ramanujan Photo Gallery, with 110 photographs on the life and work of the genius, was later set up by Srinivasa Rao, next to the Gallery. Today the museum display includes Ramanujan’s original passport, facsimiles of his notebooks and the bronze bust of Ramanujan by Paul Granlund that was presented to Janaki. Another statue of Ramanujan sculpted by K.G. Ravi was installed at the entrance of the Operational Head Quarters of the Society for Electronic Transactions and Security (SETS), Taramani, Chennai, in June 2010. Srinivasa Rao’s pilot multimedia project on the Life and Work of Srinivasa Ramanujan, made in 1999 for the Indian Science Congress Exhibition in Chennai, finally bloomed into two full-length CD-ROMs. The Institute of Mathematical Sciences (IMSc), Chennai and C-DAC’s (Centre for Development of Advanced Computing) National Multimedia Resource Centre, Pune, were sponsored by DST, Government of India, to produce these CDs. The audiobook version of David Leavitt’s The Indian Clerk narrated by Graeme Malcolm was published in 2009 by Audible Modern Vanguard. The Ramanujan Math Park, located in Kuppam, Andhra Pradesh, a joint project of Agastya International Foundation and the non-profit organization Gyanome, was conceived, partially funded and executed by Sujatha Ramadorai and her husband Srinivasan Ramadorai along with V.S.S. Sastry, an Indian mathematics communicator. Inaugurated on 22 December 2017, it is a museum and activity centre dedicated to mathematics education through hands-on teaching methods for school children. The park features both indoor and outdoor exhibits as well as interactive touch screen stations, all designed to enhance the mathematical experience. More recently, in March 2019, eminent mathematician V.S. Varadarajan and his wife Veda donated $ 1 million to establish the Ramanujan Visiting Professorship in the Mathematics Department of UCLA.

News report when Ramanujan went missing from home, 1905 *The Hindu*

Turning to the cultural impact of the story of Ramanujan’s life on society as a whole, it is quite amazing to note that, time and again the almost unbelievable and romantic journey of Ramanujan against all odds, his ultimate success reaching the highest echelons of international mathematical fame and tragic premature death has drawn the attention of the storytellers of all varieties, be it a novel, a drama or on celluloid. American playwright David Freeman came to India during the height of the colourful centenary celebrations to mark Ramanujan’s birth, that inspired A First Class Man, a crowd-pleasing drama which looks at Ramanujan’s relationship with G.H. Hardy. It had a successful off-Broadway premiere in 2006, staged by the Alter Ego theatre company. The play deconstructs Ramanujan’s Cambridge years, touching on the loneliness of an outsider trapped in Cambridge University’s elitist and very closed society. British Canadian film director Roger Spottiswoode, of the James Bond movie Tomorrow Never Dies fame, had picked up the film rights to this drama in 2011. Spottiswoode relied on multiple sources for the film, including a sizable chunk from Ramanujan: the Man and the Mathematician by S.R. Ranganathan. He said that the “script concentrates largely on the time Ramanujan was at Cambridge; so it deals closely with his relationship with two other Trinity mathematicians: Hardy and Littlewood. It is the story of the friendship between these three and, although they discuss their work, it is a story of relationships and not of mathematical equations.” In early 2010, Bollywood director Rohit Jugraj announced he would make a biopic as well. However, as of May 2020, both projects are yet to see the light of day. In 2007, English playwright Simon McBurney conceived and directed a brilliant play inspired by the collaboration between Ramanujan and Cambridge don G.H. Hardy, entitled A Disappearing Number co-written and devised by the Thé&‌acirc;tre de Complicité company. After a successful international tour, this mesmerizing performance was staged in India during the 2010 ICM (International Congress of Mathematicians) at Hyderabad.

News of Ramanujan’s departure to England *The Hindu*

Incidentally, the logo of ICM 2010, the first-ever mathematical event of that scale to be held in India, depicted the statement of the Ramanujan Conjecture or the so-called Tau conjecture that Ramanujan proposed in a 1916 paper “On Certain Arithmetical Functions”. A beautiful docu-feature film on Ramanujan was made several years ago by Nandan Kudhyadi under the auspices of National Film Development Corporation (NFDC), India. Bollywood actors Raghubir Yadav and Tom Alter played the roles of Ramanujan and Hardy respectively. In 2012–13, Nandan directed another documentary on Ramanujan, entitled The Genius of Srinivasa Ramanujan. Developed by IISER Pune, this documentary is presented mostly through mutual conversations of many notable Ramanujan scholars of present-day, though anchored mainly by A. Raghuram. Meanwhile, a Bollywood biopic Ramanujan, shot back-to-back in English and Tamil, on the inspiring life of Ramanujan was released in India on 11 July 2014. Directed by the national award-winning director Gnana Rajasekharan, the central cast of Ramanujan was Abhinay Vaddi, a grandson of legendary Gemini Ganesan, while Suhasini Maniratnam played the role of his mother Komalatammal. On 17 September 2015, a Hollywood film based on Kanigel’s book and of the same title as the book itself was finally released after almost ten years that it took to be created. Produced by Edward R. Pressman, adapted and directed by Matthew Brown, the filmmakers engaged mathematicians Ken Ono and Manjul Bhargava to collaborate on the content of the film. The movie is praised by mathematicians and scientists for its accurate portrayal of mathematics and mathematicians. It was screened at several international film festivals and became quite popular among common folk and mathematics enthusiasts alike. Popular Bollywood actor Madhavan, best known for Three Idiots, was initially chosen to play the role of Ramanujan. However, Ramanujan was finally portrayed by Dev Patel, the hero of the Oscar-winning film Slumdog Millionaire, while Jeremy Irons played Hardy.

Part II

Let us now turn to the unbelievable legacy of mathematics that Ramanujan has left behind and the following research-related developments. At the Harvard-MIT Current Developments in Mathematics Conference 2008, Ken Ono, the Asa Griggs Candler Professor of Mathematics and Computer Science at Emory University, and a noted Ramanujan scholar pointed out that “The legend of Ramanujan has continued to grow with the ever-increasing importance of his mathematics.” Shortly after Ramanujan’s death, Hardy wrote,

Opinions may differ about the importance of Ramanujan’s work, the kind of standard by which it should be judged, and the influence which it is likely to have on the mathematics of the future…. He would probably have been a greater mathematician if he could have been caught and tamed a little in his youth. On the other hand, he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain…

Ono emphatically claims, “In view of the last eighty five years of progress in number theory, it is clear that the loss would have been much greater than the gain.” Analyzing his claim, he points out,

on one hand, as Hardy did, we may largely base our conclusion on the contents of Ramanujan’s notebooks, which, apart from the “lost notebook” that contained the work of his last year, were known to mathematicians at the time of his death. They are a repository of thousands of cryptic entries on evaluations and identities of strangely named functions. Through the tireless efforts of B.C. Berndt, adding to the accumulated effort of earlier mathematicians such as Hardy, G.N. Watson, B.M. Wilson, and R.A. Rankin, a clear picture has emerged which reveals Ramanujan’s incredible gift for formulas and combinatorial relations. Perhaps he was the greatest such mathematician of all time. On the other hand, this flattering assessment is grossly inadequate, for it does not take into account Ramanujan’s impact on Hardy’s “mathematics of the future”. Indeed, number theory has undergone a tremendous evolution since Ramanujan’s death, and today it bears no resemblance to the number theory of his day. The subject is now dominated by the arithmetic and analytic theory of automorphic and modular forms, the study of Diophantine questions under the rubric of arithmetical algebraic geometry, and the emergence of computational number theory and its applications. These subjects boast many of the most celebrated achievements of 20th century mathematics such as Deligne’s proof of the Weil Conjectures, the solution to Gauss’ Class Number Problem by Goldfeld, Gross, and Zagier, Wiles’ proof of Fermat’s Last Theorem, and Borcherds’s work on the infinite product expansions of automorphic forms. A proper assessment of Ramanujan’s greatness must then take into account the remarkable fact that his work, the portion which was known to Hardy, makes intimate contact with all of these notable achievements. Clearly, Ramanujan was a great anticipator.¹ His work provided examples of deeper structures and suggested important questions which are now inescapable in the panorama of modern number theory.

Then turning towards the contents of the “lost” notebook, the mathematics that Ramanujan created on his deathbed, of which Hardy was unaware at the time of Ramanujan’s death aside from the January 20, 1920 letter on the `mock theta’ function that he had received from Ramanujan, Ono dramatically posed the question, “What are the secrets of the mathematical scrawl Ramanujan penned during his last days? What is its impact on Hardy’s mathematics of the future?” The answer to this question shapes the core of the hundred-odd pages of his masterly exposé “Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory” where he begins:

Modular forms are central in contemporary mathematics. Indeed, modular forms play crucial roles in algebraic number theory, algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics. The recent history of the subject includes (to name a few) great successes on the Birch and Swinnerton–Dyer Conjecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat’s Last Theorem. These celebrated works are dramatic examples of the evolution of mathematics; indeed, it would have been impossible to prophesy them fifty years ago. Instead of travelling back in time to the 1950s, our story begins in 1887, in a village in India. Our mathematics, which is about harmonic Maass forms, begins with the legend of the great mathematician Srinivasa Ramanujan, and the mathematics he conjured from his deathbed.

Poster of the ICTS Srinivasa Ramanujan Lecture Series ICTS

Trying to decipher what might have motivated Ramanujan to cook up `mock theta’ functions, Ono suggested, “it is not difficult to imagine Ramanujan’s mindset. It seems that Ramanujan, largely motivated by his work on partitions and the Rogers–Ramanujan identities, spent the last year of his life thinking deeply about the “near” modularity of Eulerian series. He understood the importance of developing a “new theory”, one which overlaps in spots with the classical theory of modular forms. He discovered the mock theta functions.” George Andrews’ accidental discovery of the “lost notebook” in 1976 sparked off, almost immediately, a flurry of research on the mock theta functions.

Andrews himself was a forerunner and there were many other notable names. By the late 1990s, works by Andrews, Y.-S. Choi, H. Cohen, F. Dyson, Garvan, B. Gordon, Hickerson, R. McIntosh, M. Wakimoto among numerous others, revealed many of the deeper properties of the mock theta functions. Due to the works done by them, and too many others to list, Ramanujan’s 22 mock theta functions had been related to a surprising collection of subjects: Artin L-functions in number theory, Hypergeometric functions, Partitions, Lie theory, Mordell integrals, Modular forms, Polymer Chemistry etc. In particular, Hypergeometric functions in the context of Ramanujan’s derivative formula have been explored by Balasubramanian and others. At one stage of this global development, the truth of the surprising identities, known as Mock Theta Conjecture, directly related mock theta functions to modular forms. These clues from the “lost notebook” finally placed Ramanujan’s mock theta functions in the vicinity of the theory of modular forms. Unfortunately, these clues were not enough to re-construct Ramanujan’s theory. In the language of Ono,

[D]espite a flurry of activity, the essence of Ramanujan’s theory remained a mystery. The puzzle of his last letter to Hardy, thanks to the “lost notebook,” had morphed into the enigmatic web of Ramanujan’s 22 mock theta functions. The presence of this web strongly suggested the existence of a theory, and it also demanded a solution.

In 1987, the University of Illinois at Urbana-Champaign organized the Ramanujan Centenary Conference. There, in his plenary address, Freeman Dyson beautifully summed up the situation related to mock theta functions:

The mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms which Hecke built around the old theta-functions of Jacobi. This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include mock theta-functions…. But before this can happen, the purely mathematical exploration of the mock-modular forms and their mock-symmetries must be carried a great deal further.

By the late 1990s, the vast literature on Ramanujan’s mock theta functions contained many important clues for Dyson’s “challenge for the future”. In addition to the identities comprising the mock theta conjectures, there were further clues such as $q$ -series identities relating mock theta functions to Lambert-type series and indefinite theta series. Some such identities served as motivation for the 2002 Ph.D. thesis of S. Zwegers, written under the supervision of Don Zagier.²minaire Bourbaki lecture on these relatively recent works on Ramanujan’s mock theta functions in 2007.} Indeed, Zwegers researched the following two questions of Zagier:

How do the mock theta-functions fit in the theory of modular forms?
Is there a theory of indefinite theta functions?

The real analytic modular forms of Zwegers turned out to be examples of harmonic Maass forms which were defined about the same time by Bruinier and Funke, a coincidence which catalyzed much of the contemporary research in this area. These developments opened floodgates in a wide number of new directions. Indeed, recent works by Ono, Andrews, Eguchi, Hikami, Kac, Lawrence, Malmendier, Mellit, Okada, Wakimoto, and Zagier apply this theory to areas like Donaldson invariants, gauge theory, representation theory of Lie superalgebras, knot theory, mathematical physics, probability theory and topology. Ono and his collaborators had taken up the investigation towards the answers to deep questions about many of the number-theoretic topics captured by the web of Ramanujan’s mock theta functions. In the article mentioned above, Ono described the implications of this theory to partitions and q-series, modular forms, traces of singular moduli, Borcherds products, modular L-functions `a la Kohnen–Waldspurger’ and Kohnen–Zagier. In the conclusion of his article, Ono remarked:

Although the mock theta functions are humble in origin, they have earned a distinguished role in the legend of Ramanujan. Andrews and Berndt confirm this in their article “Your hit parade: the top ten most fascinating formulas in Ramanujan’s lost notebook”. In their amusing informal poll, Ramanujan’s work on Dyson’s ranks³ and the mock theta functions rank first and second! Based on the mathematics born out of these works, as described here, it is a safe bet that ranks and mock theta functions will continue to hold these top spots into the foreseeable future.… Although Ramanujan’s last works provided the first examples of such [harmonic Maass] forms, his untimely death and the enigmatic nature of his writings resulted in a great mystery. We will never know how he came up with the mock theta functions. We certainly cannot pretend to know what he fully intended to do with them. However, it is clear that he understood that the mock theta functions would go on to play important roles in number theory, his “visions”.

How have the scholars recently tried to untangle the mythical enigma associated with the Ramanujan’s creative genius? Bruce Berndt, the mathematician who has proved each of the 3,542 theorems of the Ramanujan Notebooks, a man whose 20 years of relentless research on the three notebooks of Ramanujan has been compiled into five volumes, has a pensive take on this issue. In an interview given in Chennai to Frontline in August 1999, he said, “Many people falsely promulgate mystical powers to Ramanujan’s mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks.” According to him, the absence of proofs was “perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash.” Upon being asked about any serious error made by Ramanujan in the notebooks, he said, “there are a number of misprints. I did not count the number of serious mistakes but it is an extremely small number – maybe five or ten out of over 3,000 results. Considering that Ramanujan did not have any rigorous training, it is really amazing that he made so few mistakes.” Besides these five volumes, Berndt has written over 100 papers on Ramanujan’s works and along with George Andrews, he authored five more volumes of Ramanujan’s Lost Notebooks, published by Springer, the latest one being in 2018. When asked about the nature of Ramanujan’s mathematics in the notebooks, Berndt pointed out that,

Bruce Berndt with the slate on which Ramanujan scribbled his thoughts Bruce Berndt

To much of the mathematical world and to the public in general, Ramanujan is known as a number theorist. Hardy was a number theorist, but he was also into analysis. When Ramanujan was at Cambridge with Hardy, he was naturally influenced by Hardy. And so most of the papers he published while he was in England were in number theory. His real great discoveries are in partition functions. Along with Hardy, he found a new area in mathematics called probabilistic number theory, which is still expanding. Ramanujan also wrote sequels to his papers on highly composite numbers and arithmetical functions. There are half a dozen or more of these papers that made Ramanujan very famous. They are still very important papers in number theory. However, the notebooks do not contain much of number theory. It is, broadly speaking, in analysis. I will try and break that down a little bit. I would say that the area in which Ramanujan spent most of his time, more than any other, is in elliptic functions (theta functions), which have strong connections with number theory. In particular, Chapters 16 to 21 of the second notebook and most of the unorganized portions of the notebooks are on theta functions. There is a certain type of theta functions identity which has applications in other areas of mathematics, particularly in number theory, called modular equations. Ramanujan devoted an enormous amount of effort on refining modular equations.

In recent years some remarkable work has been done in geometric function theory where modular equations have played an important role. Indeed, the so-called extremal distortion functions have been defined in terms of solutions of modular equations. Examples of such functions are the distortion function of the quasiconformal analogue of the Schwarz lemma and the extremal distortion function of Schotty’s theorem for analytic function. These can be found in the works of G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen and others.

In 1913, Ramanujan submitted his First Quarterly Report to the Board of Studies of the University of Madras. As per Ramanujan’s Notebooks (Part I, page 295) by Bruce Berndt, these reports have never been published. Among other formulae, it contained the beautiful theorem, as usual without formal proof, which came to be known as `Ramanujan’s Master Theorem’. However, Hardy presented this in his book on Ramanujan’s work and provided a rigorous proof of it for a natural class of functions and a natural set of parameters by means of the Residue Theorem in complex analysis. Also in 1937, after Ramanujan’s death, Hardy published a paper relating Ramanujan’s work to Fourier transforms. In 1997, Wolfgang Bertram of the Institut für Mathematik, Germany, proved a version of Ramanujan’s Master Theorem where the multiplicative group of positive reals was replaced by certain Riemannian manifolds with a large group of symmetry. In 2012, Gestur Ólafsson and Angela Pasquale, further generalized the result of Bertram for a more general class of manifolds.

Original manuscripts of Ramanujan *The Hindu*

There have been many works on Ramanujan’s ground-breaking discoveries on the congruences of partition function $p(n)$ by G.E. Andrews, A.O.L. Atkin, F. Dyson and many others. Ramanujan conjectured (and in some cases proved) that there are further congruence properties in which the moduli are powers of 5, 7, or 11. Subsequent works by Atkin and Watson resolved the conjectured congruences of Ramanujan, and in an important paper Atkin somewhat experimentally discovered completely new congruences modulo some further small primes. Meanwhile, attacking the problem in a more systematic manner and using the theory of modular forms, Ono found analytically Atkin-type congruences for all prime moduli exceeding 3, and in a relatively recent work of K. Ono and S. Ahlgren in 2001, they have further extended these results to include all moduli co-prime to 6. In particular, it turns out that there are such Ramanujan-type congruences for every modulus co-prime to 6. This comprehensive theory relies on the deep works of Deligne, Serre, and Shimura. Ramanujan’s work on $p(n)$ , and the research it inspired, according to Ono, “underscores the fact that the theory of partitions has historically served as a delightful `testing ground’ for some of the deepest developments in the theory of modular forms” which includes the “interplay between the Deligne–Serre theory of $\ell$ -adic Galois representations, the `language’ of the proof of Fermat’s Last Theorem, and Shimura’s theory of half-integral weight modular forms.”

Ramanujan’s conjecture on the size of the tau function was recast in the language of representation theory by I. Satake in the 1960s. This generalized Ramanujan conjecture occupies a central position in the theory of automorphic representations and has ramifications that are crucial in diverse areas of pure as well as applied mathematics. For example, an analogue of the Ramanujan conjecture in the context of graphs (Ramanujan graph) is used in the construction of high-speed communication networks. A fundamental contribution to Ramanujan Graphs was made by A. Lubotzky, R. Phillips and P. Sarnak. Till date, the strongest result towards the generalized Ramanujan conjecture is due to H. Kim and P. Sarnak.

Ramanujan’s congruences for the tau function had inspired Jean-Pierre Serre, P. Deligne, H.P.F. Swinnerton-Dyer, and several other great mathematicians of the twentieth century to develop a deep theory connecting modular forms to Galois representations. Serre, the youngest ever recipient of the Fields Medal and the inaugural Abel Prize as well, made a famous conjecture in this theory dating back sometime in the 1970s. It was solved in complete generality by Chandrasekhar Khare and Jean-Pierre Wintenberger in 2008. They were awarded the prestigious Cole Prize for Number Theory in 2011 for this achievement.

Ramanujan’s candidature put up for election to FRS. Writing of the main text is that of Hardy. Among the signatories `From Personal Knowledge’ are E.W. Hobson and H.J. Baker, to whom Ramanujan had sent his results before he wrote to Hardy in January 1913. The Royal Society

There have been beautiful works carried out by J.H. Conway, W.A. Schneeberger, Manjul Bhargava and J. Hanke in relation to Ramanujan’s work on universal quaternary forms and his question of determining all universal quadratic forms. In 1993, Conway and Schneeberger announced a proof of the “15 theorem” which states that if a quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. But the proof was complicated and was never published. Manjul Bhargava, who is presently the R. Brandon Fradd Professor of Mathematics at Princeton University, and also holds Adjunct Professorships at TIFR, IIT Mumbai, and the University of Hyderabad, found a much simpler proof which was published in 2000. In 2005, Manjul Bhargava and Hanke announced a proof of Conway’s conjecture that a similar theorem holds for quadratic forms with integer coefficients, with the constant 15 replaced by 290. Incidentally, Manjul Bhargava was lauded with the Fields Medal in 2014 for his works related to the geometry of numbers.

The classical Rogers–Ramanujan identities continue to be studied by researchers in various fields that include number theory, modular forms, representation theory of Lie algebras and combinatorics. Among many important works in this area, recently D.S. Lubinsky used these identities to disprove a conjecture due to G.A. Baker, J. L. Gammel, and J.G. Wills from 1961 in the theory of Padé approximation which deals with the fast approximation of analytic functions by rational functions, generalizing the notion of Taylor expansion of smooth functions which deals with polynomial approximations.

An odyssey of recent developments on Ramanujan’s mathematics remains incomplete unless one points out the tremendous influence of the `circle method’ introduced by Hardy and Ramanujan in 1918. It has been applied in hundreds of papers and has been refined and modified in many directions and it is still very much in use today. In particular, Hardy and Littlewood were the first to develop the method, followed by Vinogradov, Lou Keng Hua and Kloosterman, who are among the major contributors. Traditionally its main use was limited to additive problems in number theory and Diophantine questions such as the Waring problem or the Goldbach problem. In recent times one may find them in the works of Robert Vaughan, Trevor Wooley and Kevin Ford among many others. However, this powerful method has found applications in diverse areas of mathematics. Some of the major applications are being found in combinatorics in the works of J. Bourgain, B. Green among others; analytic theory of modular forms and $L$ -functions in the works of H. Iwaniec, M. Jutila and R. Munshi among others, arithmetic algebraic geometry in the works of R. Heath-Brown, T.D. Browning and others; arithmetic in function fields in the works by Yu-Ru. Liu and T.D. Wooley and also in harmonic analysis in the works by E.M. Stein, S. Wainger, A. Magyar and others. A comparatively recent result using circle method has been announced by J. Bourgain and A. Kontorovich, that makes great progress towards proving a 1971 conjecture of Zaremba predicting that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant.

His famous image from the Passport The Royal Society

To celebrate the centenary of Srinivasa Ramanujan’s election as an FRS (that happened on 2 May 1918), Ken Ono, along with George Andrews, Manjul Bhargava and Robert Vaughan, organized a public discussion meeting at which leading scientists spoke about Ramanujan’s legacy to mathematics and science. This meeting was held on 15 and 16 October 2018 at Carlton House, London. Fifteen distinguished scientists spoke about Ramanujan’s mathematics and his extraordinary legacy across computer science, electrical engineering, mathematics and physics. The lectures described Ramanujan’s influence on a wide variety of modern topics. Apart from Ramanujan’s mock theta function and its influence on the present-day mock modular forms and quantum modular forms, these lectures also included topics like sphere packing, the impact of the Ramanujan sum in digital signal processing, the influence of his original work on $q$ -series, growth of coefficients of modular forms and mock modular forms on supersymmetry of string theory, black hole entropy and `moonshine’, a mysterious subject that seems to lie at the boundary of physics and mathematics.

So much for an utterly incomplete highlight of the mathematical developments related to Ramanuajn’s works that took place through the last few decades,⁴ particularly since the publication of the book by Kanigel in 1991. However, another relatively recent interesting development is noteworthy. It has been reported by most of his biographers including Kanigel, Abdi and Ranganathan, that during his Cambridge days, Ramanujan was very keen to communicate to his friend P.C. Mahalanobis, a glimpse of his own philosophical theory which involved an idea of multiplication between zero and infinity, the outcome of which would be the whole set of real numbers. From the reminiscences of Mahalanobis, as recorded by Ranganathan,

Left to himself, he would often speak of certain philosophical questions. He was eager to work out a theory of reality which would be based on the fundamental concepts of “zero”, “infinity” and the set of finite numbers. I used to follow in a general way but I never clearly understood what he had in mind. He sometimes spoke of “zero” as the symbol of the absolute (Nirguna-Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words, and which is completely beyond the reach of the human mind. According to Ramanujan, the appropriate symbol was the number “zero”, which is the absolute negation of all attributes. He looked on the number “infinity” as the totality of all possibilities, which was capable of becoming manifest in reality and which was inexhaustible. According to Ramanujan, the product of infinity and zero would supply the whole set of finite numbers. Each act of creation, as far as I could understand, could be symbolized as a particular product of infinity and zero, and from each such product would emerge a particular individual of which the appropriate symbol was a particular finite number. I have put down what I remember of his views. I do not know the exact implication. He seemed to have been perhaps emotionally more interested in his philosophical ideas than in his mathematical work. He spoke with such enthusiasm about the philosophical questions that sometimes I felt he would have been better pleased to have succeeded in establishing his philosophical theories than in supplying rigorous proofs of his mathematical conjectures.

Mahalanobis could not fathom the mathematical idea behind Ramanujan’s philosophical thinking, which seemed untenable in our usual set up of a number field, nor could any Ramanujan scholar since then interpret it, while some authors like Abdi, have even trodden an oversimplified path to profess that, perhaps Ramanujan had thought that any finite number divided by zero was infinity, whence he had concluded that zero multiplied by infinity would be any finite number. However, when in 2015, Avinash Sathaye, a professor of Mathematics at the University of Kentucky came forward with a paper entitled “Bhāskarācārya’s Treatment of the Concept of Infinity”, published in Gaṇita Bhāratī, it was immediate that, when Sathaye’s interpretation of the concepts of khahara and khaguna as per Bhāskarācārya is seen in the light of `idempotence’ sitting at the heart of the new algebraic structure proposed by Sathaye, not only the sums of Bījagaṇita that were so long being considered to be “wrongly framed” by Bhāskarācārya, got commensurately interpreted in the new structure, but also this new algebraic structure perfectly matches Ramanujan’s apparently bizarre mathematical idea, that came to us through the reminiscence of Mahalanobis.

Stamp on Srinivasa Ramanujan commemorating December 22 as the National Mathematics Day and 2012 as the National Mathematics Year. Wikimedia Commons

Ramanujan once told Janaki that his `name will live for one hundred years’. How amazingly right he was! On this year of his death centenary, his name appears in the titles of over 1500 research articles in mathematics, as a quick search in MathSciNet reveals, of which 10 are from 2020 itself. This numeral arguably may not be proportional to his greatness, but this may still be regarded as some measure to judge the relevance of his beautiful mathematics even in the present century.

To commemorate his death centenary this year, Vigyan Prasar, an autonomous body under DST, Govt. of India, led by its director Nakul Parashar, decided to take up what they called `Ramanujan Yatra’, a large-scale mathematics awareness and popularization campaign for high school and college students, mathematics teachers, communicators and enthusiasts, covering every direction, nook and corner of India. Towards organizing the events in a pan-India fashion, a National Core Committee was formed with T.V. Venkateswaran as its national coordinator. In its first meeting held on 11 February 2020 at New Delhi, the committee chalked out an eight-month-long program of different series of lecture demonstrations by eminent scholars, math-science fairs and expos. These were planned to be held once every month as a two/three-day program in different cities across India, to be inaugurated on 26 April 2020, the day of his untimely sad demise, with a befitting program at Kumbakonam, the place where Ramanujan was brought up and to culminate the `Yatra’ with a final program on 22 December, the National Mathematics Day, at New Delhi. But, due to the outbreak of the COVID-19 pandemic and the subsequent nationwide lockdown, the initial program at Kumbakonam had to be organized locally in a very low-key manner while maintaining physical distance with a handful people, at the Town High School campus and also at Government Arts College campus, where Ramanujan studied. As this article is going to press, the Core Committee has reformulated their plan to initially start with holding webinars across the country in the coming months. Ramanujan left us exactly 100 years ago. But he still lives and will continue to live through his immortal work. Indeed, the impact of the story of his life as a fountainhead of inspiration to society as a whole seems to be growing day by day.

Ramanujans never die.

acknowledgement I am grateful to Professors M.S. Raghunathan, C.S. Aravinda, M.K. Sen and Mahan Maharaj for their kind encouragement during my penning down of the original article back in 2012. Cordial thanks are due to dear friends Professors Swagato K. Ray and Satadal Ganguly of ISI, Kolkata. Heartfelt thanks and gratitude to medical persons Dr. Arunaloke Bhattacharya and Dr. D.N. Guha Majumdar for their valuable help.

References

W.H. Abdi, Toils and Triumphs of Srinivasa Ramanujan: The Man and the Mathematician, National, 1992.
B.C. Berndt, “Rediscovering Ramanujan”, Interview in Frontline, Vol.16, Issue-17, Aug, 14-27, 1999.
B.C. Berndt and R.A. Rankin, Ramanujan: letters and commentary, History of Mathematics, 9. Amer. Math. Soc., Providence, RI, 1995.
——- Ramanujan: Essays and Surveys, History of Mathematics, 22, Amer. Math. Soc., Providence, RI, 2001.
F. Dyson, A walk through Ramanujan’s garden, Ramanujan revisited (Urbana-Champaign, Ill. 1987), Academic Press, Boston, 1988, pages 7-28.
J.A. Harvey, “Ramanujan’s influence on string theory, black holes and moonshine”, Philos. Trans. Roy. Soc. A 378(2020), no.2163, 20180440, 13pp.
R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, Scribners,1991, Rupa 1992 (Indian edition)
Parthasarathi Mukhopadhyay, “Srinivasa Ramanujan: Transcending Kanigel’s Canvas”, Mathematics Newsletter, Ramanujan Mathematical Society, 24, no.3 December(2013), pages 63-77.
K. Ono, “Honoring a gift from Kumbakonam”, Notices Amer. Math. Soc. 53 (2006), no. 6, pages 640-651.
S. Ram, Srinivasa Ramanajuan, National Book Trust, New Delhi, 2nd ed.,2000
S. Ramaseshan, “Srinivasa Ramanujan”, Current Science 59, (1990) pages 1309-1316.
M. Ram Murty, V. Kumar Murty. The Mathematical Legacy of Srinivasa Ramanujan , Springer India, 2012
E. Shils, “Reflections on tradition, centre and periphery, and the universal validity of science: the significance of the life of S. Ramanujan”, Minerva, 29, (1991) pages 393-419.
K. Srinivasa Rao, Srinivasa Ramanujan, East West Books, Madras, 1998
K. Srinivasa Rao, “Srinivasa Ramanujan—from Kumbakonam to Cambridge”, Asia Pacific Mathematics Newsletter, Vol. 2 No. 2, World Scientific Publishing.
Avinash Sathaye, “Bhāskarācārya’s Treatment of the Concept of Infinity”, Gaṇit Bhāratī , Vol. 37, No. 1-2, (2015), pages 55-67.
P.P. Vaidyanathan, S. Tenneti, “Srinivasa Ramanujan and Signal processing problem”, Philos. Trans. Roy. Soc. A 378(2020), no.2163, 20180446, 16pp.
D.A.B. Young, “Ramanujan’s Illness”, Notes Rec. Royal Soc. London, 48 (1994), pages 107-119. $\blacksquare$

Footnotes

Ono gives the credit of such a description of Ramanujan to Manjul Bhargava, FRS, the second youngest-ever full Professor of Princeton, another major player in modern number theory. ↩
Zagier delivered a Séminaire Bourbaki lecture on these relatively recent works on Ramanujan’s mock theta functions in 2007. ↩
Although Dyson defined the notion of a partition rank in 1944, Ono thinks it is clear that Ramanujan understood the notion in 1920 because of certain identities he recorded in the “Lost Notebook”. ↩
An interested reader may find a rather long list of some relevant papers in the References of the article published in the RMS Mathematics Newsletter in its December 2013 issue. ↩

Parthasarathi Mukhopadhyay (dugumita@gmail.com) is Associate Professor at the Department of Mathematics of the Ramakrishna Mission Residential College (Autn.) at Narendrapur, Kolkata. He translated The Man Who Knew Infinity into Bengali.

This is a revised and updated version of the following article originally published in Mathematics Newsletter of the Ramanujan Mathematical Society, and has been republished here with permission: Parthasarathi Mukhopadhyay. 2013. “Srinivasa Ramanujan: Transcending Kanigel's Canvas’’, Mathematics Newsletter, 24(3): 63–77.