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Srinivasa Ramanujan: Going Strong at 125, Part I Krishnaswami Alladi, Editor

The 125th anniversary of the birth of the Indian and startling beauty of these identities convinced mathematical Srinivasa Ramanujan falls Hardy that Ramanujan was on a par with Euler and on December 22, 2012. To mark this occasion, the Jacobi in sheer manipulative ability. At Hardy’s Notices offers the following feature article, which invitation, Ramanujan went to in 1914. appears in two installments. The article contains an The rest is history. In the five years he was in introductory piece describing various major devel- , Ramanujan published several important opments in the world of Ramanujan since his centen- papers, some of which stemmed from his discover- nial in 1987, followed by seven pieces describing im- ies in India; he collaborated with G. H. Hardy and portant research advances in several areas of math- wrote two very influential papers with him. For ematics influenced by him. The first installment in his pathbreaking contributions, Ramanujan was the present issue of the Notices contains the intro- honored by being elected Fellow of Trinity College, ductory piece by Krishnaswami Alladi, plus pieces Cambridge, and Fellow of the Royal Society (FRS) by George Andrews, Bruce Berndt, and Jonathan even though he did not have a college degree! The Borwein. Pieces by , K. Soundararajan, R. C. rigors of life in England during the First World War, Vaughan, and S. Ole Warnaar will appear in the combined with his own peculiar habits, led to a second installment in the January 2013 issue of the rapid decline in his health. Ramanujan returned to Notices. Madras, India, in 1919 a very sick man and died shortly after on April 20, 1920. Even during his Krishnaswami Alladi last days, his mathematical creativity remained undiminished. He wrote one last letter to Hardy Ramanujan’s Thriving Legacy in January 1920 outlining his discovery of the Srinivasa Ramanujan is one of the greatest mathe- mock theta functions, considered to be among maticians in history and one of the most romantic his deepest contributions. In the decades after figures in the mathematical world as well. Born Ramanujan’s death, we have come to realize the on December 22, 1887, to a poor Hindu depth, breadth, and significance of his many discov- family in in the state of in eries. Ramanujan’s work has had a major influence South India, Ramanujan was a self-taught genius on several branches of , most notably who discovered a variety of bewildering identities in and classical analysis, and even starting from his school days. There is a legend that in some areas of physics. goddess Namagiri in the neighboring During the Ramanujan Centennial in 1987, town of used to come in his dreams and eminent from around the world give him these formulae. Unable to find individuals gathered in India to pay homage to this singular in India to understand and evaluate his findings, genius. It was an appropriate time to reflect on his Ramanujan wrote two letters in 1913 to G. H. legacy and consider the directions in which his work Hardy of Cambridge University, England, listing would have influence in the future. In this quarter several of his most appealing formulae. The depth century since the centennial, many significant developments have taken place in the world of Krishnaswami Alladi is professor of mathematics at the Uni- Ramanujan: (i) there have been major research versity of Florida, Gainesville. His email address is alladik@ advances in recent years that have widened the ufl.edu. arena of impact of Ramanujan’s work; (ii) books DOI: http://dx.doi.org/10.1090/noti917 on Ramanujan’s work, of interest to students

1522 Notices of the AMS Volume 59, Number 11 and researchers alike, and books on Ramanujan’s life with appeal to the general public have been published; (iii) journals bearing Ramanujan’s name have been established, one of which is devoted to all areas of research influenced by him; (iv) annual conferences devoted to various aspects of Ramanujan’s work are being held in his hometown in India; (v) international prizes bearing his name and encouraging young mathematicians have been created; and (vi) movies and plays about Ramanujan inspired by his remarkable life have also appeared or are in production. This feature in the Notices of the AMS is devoted to describing some of the significant developments in the world of Ramanujan since the centennial. My opening article will broadly address the devel- opments listed under (i)–(vi). This will be followed by articles by leading mathematicians on various aspects of research related to Ramanujan’s work.

Some Major Research Advances I focus here on certain aspects of the theory of partitions and q-hypergeometric with which I am most familiar and provide just a sample of some very important recent results. Research relating to Ramanujan’s work in many other areas

has been very significant, and some of these ideas Photo courtesy of Krishnaswami Alladi. will be described by others in the following articles. The bust of Ramanujan now stands in the central Ramanujan’s mock theta functions are mysteri- hall of his home in . The bust was ous, and the identities he communicated to Hardy added after SASTRA University purchased about them are deep. Since George Andrews un- Ramanujan’s home. In the background is a earthed the Lost Notebook of Ramanujan at Trinity replica of his famous passport photograph. College Library in Cambridge University in 1976, he has been analyzing the identities contained therein, especially those on the mock theta func- 11, where p(n) is the number of partitions of tions. We owe much to Andrews for explaining n. In the ensuing decades, this led to a fruitful the mock theta function identities in the context study of congruences satisfied by coefficients of of the theory partitions (see for instance [12]). In modular forms. Yet it was unclear whether the Andrews-Garvan [21] the mock theta conjectures partition function itself satisfied congruences with are formulated. These relate the q-hypergeometric respect to other primes. Oliver Atkin [22] was mock theta function identities to partition identi- the first to discover congruences for the partition ties; the mock theta conjectures were proved by function of the type p(An + b) ≡ C(mod p) with a Hickerson [40] around the time of the centennial. prime modulus p > 11, but unlike the Ramanujan In spite of their resemblance to the classical congruences, we have A not equal to p. In recent theta functions, the exact relationship between the years Scott Ahlgren [1] and Ken Ono [44] have mock theta functions and theta functions remained discovered partition congruences to larger prime unclear. In the last decade, dramatic progress on moduli and have shown that there are indeed this matter has been achieved by Ken Ono and infinitely many such congruences. his coworkers, most notably Kathrin Bringmann, Ramanujan (see [46]) wrote down two spec- in establishing the links between mock theta tacular identities for the generating functions of functions and Maass forms, thereby providing the p(5n + 4) and p(7n + 5), from which his congru- key to unlock this mystery [32], [33], [34]. ences modulo 5 and 7 follow. Proofs of Ramanujan’s Ramanujan’s congruences for the partition func- partition congruences were given by Atkin using tion modulo the primes 5, 7, and 11 and their the theory of modular forms. In 1944, Freeman powers are among his most significant discov- Dyson [35] gave a combinatorial explanation of eries. More specifically, Ramanujan discovered Ramanujan’s partition congruences mod 5 and 7 that p(5n + 4) is a multiple of 5, p(7n + 5) is using the now famous statistic he called the rank. a multiple of 7, and p(11n + 6) is a multiple of He then conjectured the existence of a statistic

December 2012 Notices of the AMS 1523 he dubbed the crank that would explain the mod of integrable models in conformal field theory. 11 congruence. A combinatorial explanation of For this work McCoy was recognized with the Ramanujan’s partition congruence modulo 11 was Heineman Prize in Mathematical Physics and an first given in the Ph.D. thesis of Frank Garvan [36] invited talk at the ICM in Berlin in 1998. using vector partitions and a vector crank. This was My own work in the theory of partitions and then converted to ordinary partitions by Andrews q-hypergeometric series has been in the study and Garvan [20] when they got together at the Ra- of R-R type identities and their refinements [4], manujan Centennial Conference at the University especially on a partition of Göllnitz, one of Illinois, Urbana, in the summer of 1987, and so of the deepest R-R type identities. In collaboration finally the crank conjectured by Dyson was found. with Basil Gordon, George Andrews, and Alexander Perhaps the Goddess of Namakkal made sure Berkovich, I was able to obtain generalizations that the solution to this problem should actually and refinements of Göllnitz’s partition theorem happen during the Ramanujan Centennial! Since in 1995 [9] and later, in 2003, a much deeper 1987 there has been an enormous amount of work four-parameter extension of the Göllnitz theorem on the rank and on cranks for various partition sta- [10]. tistics by several mathematicians: Garvan, Kim, and Stanton [37] found cranks for t-cores; Mahlburg Books on Ramanujan’s Life and Work [43] found cranks for the partition congruences Hardy’s twelve lectures on Ramanujan [39] are to moduli larger than 11; and Andrews [16] has a classic in exposition. For several decades after noticed new connections with Durfee symbols of Ramanujan’s death, this book and the Collected partitions. In the last decade, Richard Stanley [49] Papers [46] of Ramanujan were the primary sources introduced a refinement of Dyson’s rank, and this for students and young researchers aspiring to has led to important refinements and extensions enter the Ramanujan mathematical garden. In 1987, of Ramanujan’s partition congruences (see for The Lost Notebook of Ramanujan was published instance Andrews [15] and Berkovich-Garvan [24]). [48]. Students today are very fortunate, because In the entire theory of partitions and Bruce Berndt has edited the original three note- q-hypergeometric series, the Rogers-Ramanujan books of Ramanujan and published this material identities are unmatched in simplicity of form, ele- in five volumes [29]. In these volumes one can find gance, and depth. This pair of identities connects proofs of Ramanujan’s and how they are partitions into parts that differ by at least two related to past and contemporary research. For this with partitions into parts congruent to ±i(mod 5), monumental contribution, Berndt was recognized for i = 1, 2. Nowadays, by a Rogers-Ramanujan with the AMS Steele Prize. In the past five years, (R-R) type identity, we mean an identity which Bruce Berndt and George Andrews have joined in its analytic form equates a q-hypergeometric forces to edit Ramanujan’s Lost Notebook. This is series to a product, where the series represents expected to also require five volumes, of which the generating function of partitions whose parts three have already appeared [19]. satisfy gap conditions and the product is the Ramanujan is now making an impact beyond generating function of partitions whose parts mathematics into society in general around the satisfy congruence conditions. This subject has world. Throughout India, Ramanujan’s life story blossomed into an active area of research thanks is well known, and Ramanujan is a hero to every to systematic analysis of R-R type identities, Indian student. But with the publication of Robert primarily by Andrews ([11], Chapter 7). In the Kanigel’s book The Man Who Knew Infinity [41], 1980s the Australian physicist Rodney Baxter Ramanujan’s life story has reached the world over. showed how certain R-R type identities arise as The impact of this book cannot be underestimated solutions of some models in statistical mechanics (see [2]). Subsequently, Bruce Berndt and Robert (see Andrews [13], Chapter 8). Soon after, Andrews, Rankin have published two wonderful books. The Baxter, and Forrester [18] determined the class first one, called Ramanujan—Letters and Com- of R-R type identities that arise in such models mentary [30], collects various letters written to, in statistical mechanics. For this work Baxter from, and about Ramanujan and makes detailed was recognized with the Boltzmann Medal of the commentaries on each letter. For instance, if a American Statistical Society. letter contains a mathematical statement, there Subsequently there has been significant work on is an explanation of the mathematics with appro- q-hypergeometric identities and integrable models priate references. If there is a statement about in conformal field theory in physics. The leader Ramanujan being elected Fellow of the Royal So- in this effort is Barry McCoy, who in the 1990s, ciety, there is a description about the procedures with his collaborators Alexander Berkovich, Anne and practices for such an election (see review [3]). Schilling, and Ole Warnaar [25], [26], [27], [28], The second book, Ramanujan—Essays and Surveys discovered new R-R type identities from the study [31], is a collection of excellent articles by various

1524 Notices of the AMS Volume 59, Number 11 experts on Ramanujan’s life and work and about various individuals who played a major role in Ramanujan’s life (see review [5]). This book also contains an article by the great , who describes how he was inspired by Ramanujan’s mathematical discoveries. Most recently, David Leavitt has published a book [42] entitled The Indian Clerk which focuses on the close relationship between Ramanujan and Hardy (see review [38]). All of these books will not only appeal to mathematicians but to students and lay persons as well. Since the Ramanujan Centennial, I have written articles annually comparing Ramanujan’s life and Photo courtesy of Krishnaswami Alladi. mathematics with those of several great mathe- Left to right: Krishnaswami Alladi, George maticians in history, such as Euler, Jacobi, Galois, Andrews (Pennsylvania State University), and Abel, and others. These articles, which were writ- SASTRA Vice-chancellor R. Sethuraman in front ten to appeal to the general public, appeared in of a bust of Srinivasa Ramanujan at SASTRA The Hindu, India’s national newspaper, each year University, Kumbakonam—December 22, 2003. around Ramanujan’s birthday in December. The collection of all these articles, along with my book reviews and other Ramanujan-related articles, has Prior to these two journals, in the 1970s Pro- just appeared as a book [8] for Ramanujan’s 125th fessor K. Ramachandra (now deceased) founded anniversary. The Hardy-Ramanujan Journal, devoted to elemen- There are many other books on Ramanujan tary and classical analytic number theory, and that have been published, and my list above is published it privately in India. certainly not exhaustive. However, I must add to this list the comprehensive CD on Ramanujan that Srinivasa Rao brought out in India a few Annual Conferences in Ramanujan’s years ago. In this CD we have scanned images Hometown of Ramanujan’s notebooks, his papers, several A recent major event in the world of Ramanujan is articles about Ramanujan, many important letters, the acquisition in 2003 of Ramanujan’s home in etc. Kumbakonam, South India, by SASTRA University. This private university, founded about twenty Journals Bearing Ramanujan’s Name years ago, has grown by leaps and bounds. Since Inspired by what I saw and heard during the Ra- the university has purchased Ramanujan’s home, manujan Centennial, I wanted to create something we have the active involvement of administrators, that would be a permanent and continuing memo- academicians, and students in the preservation of rial to Ramanujan, namely, The Ramanujan Journal, Ramanujan’s legacy. To mark the occasion of the devoted to all areas of mathematics influenced purchase of Ramanujan’s home, SASTRA Univer- by Ramanujan. This idea received enthusiastic sity conducted an international conference at its support from mathematicians worldwide, many of Kumbakonam campus December 20–22, 2003, and whom have served, or are serving, on the editorial had it inaugurated by Dr. Abdul Kalam, president board. The journal was launched in 1997 by Kluwer of India. I was invited to bring a team of math- Academic Publishers, which published four issues ematicians to this conference. George Andrews of one hundred pages per year in one volume. gave the opening lecture of this conference as well Now the journal is published by Springer, which as the concluding Ramanujan Commemoration publishes nine issues per year of one hundred fifty Lecture on December 22, Ramanujan’s birthday. At pages each in three volumes. The rapid growth of the conclusion of the conference, the participants this journal is a further testimony to the fact that suggested that SASTRA should conduct confer- Ramanujan’s work is continuing to have major ences annually in Kumbakonam on various areas influence on various branches of mathematics. of mathematics influenced by Ramanujan. I have There is another journal bearing Ramanujan’s helped SASTRA organize these annual conferences. name: The Journal of the Ramanujan Mathematical Each year several leading researchers visit SASTRA Society, which is distributed by the AMS and is in December to speak at these conferences, and devoted to all areas of mathematics. It bears Ra- by participating in them, I have had the pleasure manujan’s name since he is India’s most illustrious of spending Ramanujan’s birthday, December 22, mathematician. every year in Ramanujan’s hometown (see [6]).

December 2012 Notices of the AMS 1525 Prizes Bearing Ramanujan’s Name Finally, there is the Srinivasa Ramanujan Medal At the inauguration of the second Ramanujan given by the Indian National Science Academy conference at SASTRA University in December for outstanding contributions to mathematical 2004, Vice Chancellor Professor R. Sethuraman sciences. There is no age limit for this medal. graciously offered to set apart US$10,000 annually Since its launch in 1962, the medal has been given to recognize pathbreaking research by young fourteen times. mathematicians. In response to his offer of such support, I Movies and Plays about Ramanujan suggested that an annual prize be launched the next Ramanujan’s life story is so awe inspiring that year to be given to mathematicians not exceeding movies and plays about him have been and are being the age of thirty-two for outstanding contributions produced. The first was a superb documentary to areas influenced by Ramanujan. The age limit about Ramanujan in the famous Nova series of was set at thirty-two because Ramanujan achieved the Public Broadcasting System (PBS) on television, so much in his brief life of thirty-two years. Thus which described some of his most appealing the SASTRA Ramanujan Prize was born. In 2005, the mathematical contributions in lay terms and some year the prize was launched, the prize committee of the most startling aspects of his life, such as felt that two brilliant mathematicians, Manjul the episode of the taxi cab number 1729. This Bhargava (Princeton) and Kannan Soundararajan documentary was produced before the Ramanujan (then at Michigan, now at Stanford), both deserved Centennial but after Andrews’s discovery of the the prize for spectacular contributions to algebraic Lost Notebook. and analytic number theory respectively. The Since the Ramanujan Centennial, there have vice chancellor generously informed the prize been many stage productions, including the Opera committee that there would be two full prizes Ramanujan, which was performed in Munich in that year, and it would not be split. The prestige 1998. In 2005 I had the pleasure of seeing the of any prize is determined by the caliber of the play Partition at the Aurora Theater in Berkeley winners. The SASTRA Ramanujan Prize could not along with George Andrews. In this and other have had a better start (see [7], [45]). The prize is stage productions, there is usually a “playwright’s now one of the most prestigious and coveted. The fancy”, namely, a modification of the true life subsequent winners are: Terence Tao (UCLA) in story of Ramanujan for a special effect. The 2006, Ben Green (Cambridge University) in 2007, producer of Partition, Ira Hauptman, focuses on (Stanford) in 2008, Kathrin Ramanujan’s monumental work with Hardy on Bringmann (Cologne) in 2009, Wei Zhang (Harvard) the asymptotic formula for the partition function, in 2010, Roman Holowinsky (Ohio State) in 2011, but for a special effect describes Ramanujan’s and Zhiwei Yun (MIT and Stanford) in 2012. This attempts to solve Fermat’s Last Theorem and prize has been given in December each year in Ramanujan’s conversation with the Goddess of Ramanujan’s hometown, Kumbakonam, during the Namakkal regarding this matter. As far as we SASTRA Ramanujan Conference. know, Ramanujan never worked on Fermat’s Last There is another very prestigious prize with Theorem, but this digression for a special effect is Ramanujan’s name attached, namely, the ICTP acceptable artist’s fancy (see [8], article 23). Ramanujan Prize. The International Centre for In 2006 Andrews and I served on a panel to Theoretical Physics (ICTP) in Trieste, Italy, was critique a script entitled A First Class Man which created in the sixties by the great physicist Abdus had received support from the Sloan Foundation. Salam with a particular commitment to encour- The reading of the script was done at the Tribeca age scientists from developing countries. Keeping Film Institute in New York when the distinguished this vision of Salam, the ICTP Ramanujan Prize writer David Freeman was present. Andrews and launched in 2003 is a US$10,000 annual prize for I helped in clearing up certain mathematical mathematicians under the age of forty-five from misconceptions in the script and also noted that developing countries for outstanding research in the script had changed the Ramanujan life story in any branch of mathematics. Whereas the SASTRA certain ways that were not appropriate. Regardless Ramanujan Prize is open to mathematicians the of how the life of Ramanujan is portrayed in these world over but focuses on areas influenced by stage productions, they have had the positive effect Ramanujan, the ICTP Ramanujan Prize is open to of drawing the attention of the public around the all areas of mathematics but focuses on candidates world to the life of this unique figure in history. from developing countries. The ICTP Prize has In 2007 a play entitled A Disappearing Num- Ramanujan’s name because Ramanujan is the great- ber was conceived and directed by the English est mathematician to emerge from a developing playwright Simon McBurney for the Theatre Com- country. The ICTP Ramanujan Prize is given in plicite Company. It first played at the Theater cooperation with the Abel Foundation. Royal in Plymouth, England, and won three very

1526 Notices of the AMS Volume 59, Number 11 prestigious awards in England in 2007. This play References was performed at the International Congress of [1] S. Ahlgren and K. Ono, Congruence properties for Mathematicians in Hyderabad, India, in August the partition function, Proc. Nat. Acad. Sci. USA 98 2010. (2001), 12882–12884. [2] K. Alladi, The discovery and rediscovery of mathe- The latest theatrical production is a movie that matical genius, Book Review of The Man Who Knew is now being produced in India based on Kanigel’s Infinity (by Robert Kanigel), The American Scientist 80 book The Man Who Knew Infinity and featuring one (1992), 388–389. of the most popular actors in the Tamil cinema [3] , Book Review of Ramanujan: Letters and Com- world as Ramanujan. mentary (by Bruce C. Berndt and Robert A. Rankin), The American Math. Monthly 103 (1996), 708–713. [4] , Refinements of Rogers-Ramanujan type iden- Events in 2011–12 tities, in Proc. Fields Inst. Conf. on Special Functions, Since December 2012 is the 125th anniversary of q-Series and Related Topics, Fields Inst. Communica- Ramanujan’s birth, the entire year starting from tions, 14, Amer. Math. Soc., Providence, RI, 1997, pp. December 22, 2011, was declared as the Year of 1–35. [5] , Book Review of Ramanujan: Essays and Sur- Mathematics by the prime minister of India at a veys (by Bruce C. Berndt and Robert A. Rankin), public function in Madras on December 26, 2011. American Math. Monthly 110 (2003), 861–865. On that day, a stamp of Ramanujan was released [6] , A pilgrimage to Ramanujan’s hometown, MAA and Robert Kanigel was given a special award Focus 26 (2006), 4–6. for his biography of Ramanujan. Also, the Tata [7] , The first SASTRA Ramanujan prizes, MAA Institute released a reprinting of the Notebooks Focus 26 (2006), 7. [8] , Ramanujan’s Place in the World of Mathemat- of Ramanujan in the form of a collector’s edition ics, Springer, New Delhi, 2012, 180 pp. [47]. The mathematics year has featured several [9] K. Alladi, G. E. Andrews, and B. Gordon, General- conferences and public lectures on Ramanujan izations and refinements of a partition theorem of throughout India. Göllnitz, J. Reine Angew. Math. 460 (1995), 165–188. The Ramanujan Journal is bringing out a spe- [10] K. Alladi, G. E. Andrews, and A. Berkovich,A new four parameter q-series identity and its partition cial volume for the 125th birth anniversary in implications, Invent. Math. 153 (2003), 231–260. December 2012. The 125th anniversary celebra- [11] G. E. Andrews, The theory of partitions, in Encyl. Math., tions will conclude in 2012 with an International 2, Addison-Wesley, Reading, MA 1976. Conference on the Works of Ramanujan at the [12] , An introduction to Ramanujan’s lost notebook, University of Mysore, India, December 12–13; an Amer. Math. Monthly 86 (1979), 89–108. International Conference on the Legacy of Ra- [13] , q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics, manujan at SASTRA University in Kumbakonam and Computer Algebra, CBMS Regional Conf. Ser. in during December 14–16; and an International Con- Math., 66, Amer. Math. Soc., Providence, RI, 1986. ference on Ramanujan’s Legacy in India’s capital [14] , The hard-hexagon model and the Rogers- New Delhi during December 17–22. In 2012 only, Ramanujan identities, Proc. Nat. Acad. Sci. USA 78 the SASTRA Ramanujan Prize will be given out- (1981), 5290–5292. side of Kumbakonam, namely, at the New Delhi [15] , On a partition function of Richard Stanley, Stanley Festschrift, Elec. J. Comb. 11 (2004/06). conference. [16] , Partitions, Durfee symbols, and the Atkin- Just as there was a Ramanujan Centennial Garvan moments of ranks, Invent. Math. 169 (2007), Conference at the University of Illinois, Urbana, 37–73. in the summer of 1987 before the centennial [17] , The meaning of Ramanujan, now and for the celebrations in India that December, there was a future, Ramanujan J. 20 (2009), 257–273. [18] G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eight- Ramanujan 125 Conference at the University of vertex SOS model and generalized Rogers-Ramanujan Florida, Gainesville, in November 2012, one month type identities, J. Stat. Phys. 35 (1984), 193–266. before the 125th anniversary celebrations in India. [19] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Finally, at the Joint Annual Meetings of the AMS Notebook, Parts I–V, Springer, New York, 2005, 2009, and MAA in January 2013 in San Diego, there will 2012, to appear. be a Special Session on Ramanujan’s mathematics [20] G. E. Andrews and F. Garvan, Dyson’s , Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167– to mark his 125th birth anniversary. 171. We have much to be proud of regarding what [21] , Ramanujan’s “lost” notebook, VI: The mock has happened in the quarter century after the theta conjectures, Adv. in Math. 73 (1989), 242–255. Ramanujan Centennial. The articles in this Notices [22] A. O. L. Atkin, Multiplicative congruence properties feature convince us that the world of Ramanujan’s and density problems for p(n), Proc. London Math. Soc. 18 (1968), 563–576. mathematics is continuing to grow, and so we have much to look forward to in the next quarter century.

December 2012 Notices of the AMS 1527 [23] A. O. L. Atkin and F. Garvan, Relations between the [47] , Notebooks (2 volumes), Tata Institute of ranks and cranks of partitions, Ramanujan J. 7 (2003), Fundamental Research, Bombay, 1957 (reprinted as 343–366. Collector’s Edition in 2011). [24] A. Berkovich and F. Garvan, On the Andrews-Stanley [48] , The Lost Notebook and Other Unpublished refinement of Ramanujan’s partition congruence mod- Papers, Narosa, New Delhi, 1987. ulo 5 and generalizations, Trans. Amer. Math. Soc. 358 [49] R. Stanley, Problem 10969, Amer. Math. Monthly 109 (2006), 703–726. (2000), 760. [25] A. Berkovich and B. McCoy, Continued fractions and Fermionic representations for characters of M(p, p0) minimal models, Lett. Math. Phys. 37 (1996), 49–66. George E. Andrews [26] , Rogers-Ramanujan identities: A century of progress from mathematics to physics, Doc. Math. 1998 Partitions and the Rogers-Ramanujan Extra Vol. III ICM, 163–172. [27] A. Berkovich, B. McCoy, A. Schilling, and O. War- Identities naar, Bailey flows and Bose-Fermi identities for con- In his first letter to G. H. Hardy [19, p. xxvii], (1) (1) (1) Ramanujan asserts: formal coset models (A1 )N × (A1 )N0 /(A1 )N+N0 , Nucl. Phys. B 499 (1997), 621–649. 1 “The coefficient of xn in [28] A. Berkovich, B. McCoy, and A. Schilling, Rogers- 1 − 2x + 2x4 − 2x9 + 2x16 − · · · 0 (1) Schur-Ramanujan type identities for M(p, p ) models √ 1  √ sinh(π n)  of conformal field theory, Comm. Math. Phys. 191 = the nearest integer to cosh(π n) − √ , ” (1998), 325–395. 4n π n [29] Bruce C. Berndt, Ramanujan’s Notebooks, Parts I–V, and, in a footnote, Hardy states, “This is quite Springer, New York, 1985, 1989, 1991, 1994, 1998. untrue. But the formula is extremely interesting [30] Bruce C. Berndt and Robert A. Rankin, Ramanujan: for a variety of reasons.” Perhaps one of the most Letters and Commentary, Amer. Math. Soc., Providence, RI, and LMS, London, 1995. important justifications for calling it “extremely [31] , Ramanujan: Essays and Surveys, Amer. Math. interesting” is that it eventually led to the Hardy- Soc., Providence, RI, and LMS, London, 2001. Ramanujan formula for p(n). [32] K. Bringmann and K. Ono, The f (q) mock theta func- Each positive integer can be partitioned into tion conjecture and partition ranks, Invent. Math. 165 unordered sums of positive integers. For exam- (2006), 243–266. [33] , Dyson’s rank and Maass forms, Annals of Math. ple, there are seven partitions of the number 5: 171 (2010), 419–449. 1+1+1+1+1, 1+1+1+2, 1+2+2, 1+1+3, 2+3, 1+4, 5. [34] J. K. Bruinier and K. Ono, Heegner divisors, L- We denote by p(n) the number of partitions on n, functions, and harmonic weak Maass forms, Annals so p(5) = 7. of Math. 172 (2010), 2135–2181. Inspired by (1), Hardy and Ramanujan eventually [35] F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. proved [13] the following: [36] Francis Gerard Garvan, Generalizations of Dyson’s Theorem 1. Suppose that rank, Ph.D. Thesis, The Pennsylvania State University √ (1986), 136 pp. q d   Cλn/q φq(n) = √ e /λn , [37] Frank Garvan, Dongsu Kim, and Dennis Stanton, 2π 2 dn Cranks and t-cores, Invent. Math. 101 (1990), 1–17. [38] H. Halberstam, Review of The Indian Clerk (by David 2√π where C and λn are defined by C = and λn = Leavitt), Notices Amer. Math. Soc. 55 (2008), 952–956. q 6 − 1 [39] G. H. Hardy, Ramanujan—Twelve Lectures on his Life n 24 , and for all positive integral values of q; and Work, Cambridge University Press, Cambridge, that p is a positive integer less than and prime to 1940. q; that ωp,q is a 24q-th root of unity, defined when [40] D. Hickerson, A proof of the mock theta conjectures, p is odd by the formula Invent. Math. 94 (1988), 639–660. ! " ( ! [41] Robert Kanigel, The Man Who Knew Infinity: A Life of −q 1 1 1 ωp,q = exp − (2 − pq − p) + q − the Genius Ramanujan, Charles Scribner’s, New York, p 4 12 q 1991. ) # 0 2 0 [42] David Leavitt, The Indian Clerk, Bloomsbury, New (2p − p + p p ) πi York, 2007. [43] , Partition congruences and the and when q is odd by the formula Andrews-Garvan-Dyson crank, Proc. Nat. Acad. Sci. ! " ( ! USA 102 (2005), 15373–15376. −p 1 1 1 ωp,q = exp − (q − 1) + q − [44] K. Ono, Distribution of the partition function modulo q 4 12 q m, Annals of Math. 151 (2000), 293–307.   0 2 0 [45] , Honoring a gift from Kumbakonam, Notices (2p − p + p p ) πi , Amer. Math. Soc. 53 (2006), 640–651. [46] S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927. George E. Andrews is Evan Pugh Professor of Mathematics at The Pennsylvania State University. His email address is [email protected].

1528 Notices of the AMS Volume 59, Number 11 December surprise: major Atle further a birth, notes Ramanujan’s [22] Selberg of anniversary dredth fact, in that, prove to able now was he of definition replaced he things simplify bit, To a theorem. amazing this on notes to nearest integer the that so of part integer that and by divisible and where 2003. 20, December Kumbakonam, Street, Sannidhi home, Sarangapani Ramanujan’s Srinivasa in of Andrews front George and Alladi Krishnaswami Photo courtesy of Krishnaswami Alladi. 1 bv]wihhssm eaint his to relation some has which above] (1) nteclbain urudn h n hun- one the surrounding celebrations the In lecture preparing was Rademacher 1937 In Then n esta hsi h xc nlgeof analogue exact the is this that sees expression, that one at looks one for If coefficient. expression the approximate an as what in claims term he leading the as gives with signs It coefficients). series alternating power and of (a exponents reciprocal square series the theta certain of a coefficient the namely function, about partition the i.e., on work [ed: later there statement to a letter is first Ramanujan’s there at Hardy, looks one If p 0 (a/b) p(n) saypstv nee uhthat such integer positive any is α p(n) q 2012 A s o l ucetylrevle of values large sufficiently all for is, saypstv osat and constant; positive any is φ ; stesmo fLgnr n Jacobi and Legendre of symbol the is q that q (n) α (n) p(n) √ = n = X 2 1 1,p 9] ohsamazement, his To 699]. p. [17, ν . 1 e X (p) X ν 1 Cλ = A A q n ω q X /q φ ∞ q = p,q φ 1 q by A q + e . q − φ O(n sinh 2 npπi/q q . − (Cλ 4 1 ), ; n /q) 1 + ν nthe in pp the , 0 n, oie fteAMS the of Notices is n 1] ln ihsetclrrslso the on results spectacular of divisibility with along [10], Ono formulas for amazing is Further [25]) Vaughan. (cf. by Ramanujan discussed and Hardy by introduced olwn seto bu partitions: about the assertion implies identity following this that notes then MacMahon monumental his Analysis of VII, Chapter Section began MacMahon III, A. P. 1916 In career. Ramanu- jan’s of component surprising and intriguing Ono. by 1 + htvrwyh a bandti,hdbeen had this, obtained had he way whatever hr ntergthn ieteepnnsof exponents the side right-hand the on where eyrlvn.Itikta fHryhdtrusted had Hardy if that think I relevant. very eie sfra h offiin of coefficient the as far as verified h usqetipc fte“icemethod” “circle the of impact subsequent The that. about is doubt There little series. Rademacher the with ended inevitably have should they more, Ramanujan not as purely analyses a one for which reason, expression, technical their of in form other terms the the chose he and when insight who intuition, Ramanujan’s Hardy me trust way, fully a to not in did clear been, have seems must it it that paper, the studying expression. that of term correct the to led formula Rademacher for the in term leading the orao odutistuh u thsnot has it but established. truth; been its yet doubt to practically reason is no there that so expansion actual ≡ x relation the gives modulus I the Theorem to five. relation definite a in- have whose parts most partitions concern the these of the of teresting Two to numbers. of applications of number partition have large which a formulae suggested has bridge, an form also identities Rogers-Ramanujan The p(n) 1 r h ubr ie ytecongruences the by given numbers the are o 5 mod 1 ntewr ntepriinfunction, partition the on work the In hsms eakbetermhsbeen has theorem remarkable most This Cam- College, Trinity of Ramanujan Mr − x p(n) = x ( 1,p 3 sfollows: as 33] p. [16, × + + + aebe on yFlo,Kn,and Kent, Folsom, by found been have 1 ( ( ( ( − 1 1 1 1 AAUA IDENTITIES RAMANUJAN hc hw htRmnjn in Ramanujan, that shows which x)( p(n) − − − − , ≡ x x)( x)( x)( 1 4 x o 5 mod 4 − )( 4 1 1 1 hs set ilb discussed be will aspects these ; 1 x − − − x 6 − x )( 9 x x x i x 2 2 2 2 1 9 ( . . . ) )( ) − )( 1 1 x . − 1 11 − 1 x ( . . . ) − x 1 3 14 ) x · · · + i ( . . . ) ) 1 · · · + − 1 Combinatory x 5 − m x + x 5 1 m 89 . . . ) + 4 by . . . ) , 1529 The theorem asserts that the partitions, be no corresponding identities for moduli higher whose parts are limited to be of the forms than 5.” 5m + 1, 5m + 4, are equi-numerous with It turned out that Lehmer and Alder had those which involve neither repetitions nor shown only that the wrong generalization is the sequences. wrong generalization. The subject truly opened up Subsequently, MacMahon notes that, in the second with B. Gordon’s magnificent partition-theoretic identity, generalization [12] in 1961. x2 x6 x12 1+ + + +· · · Theorem 2. Let Bk,i(n) denote the number of par- 1 − x (1 − x)(1 − x2) (1 − x)(1 − x2)(1 − x3) + + · · · + 2 titions of n of the form b1 b2 bs , where xi +i + + · · · bj ≥ bj+1, bj − bj+k−1 ≥ 2 and at most i − 1 of the (1 − x)(1 − x2)...(1 − xi ) bj = 1. Let Ak,i(n) denote the number of partitions 1 = of n into parts 6≡ 0, ±i(mod 2k + 1). Then A (n) = (1 − x2)(1 − x7)(1 − x12)...(1 − x5m+2) k,i ≤ ≤ 1 Bk,i(n), for 1 i k. × , 3 8 13 5m+3 (1 − x )(1 − x )(1 − x )...(1 − x )... The cases k = 2, i = 1, 2 are the partition the exponents of x on the right-hand side are of theoretic consequences of the original identities the form 5m + 2 and 5m + 3. as stated by MacMahon. This relation has also been verified by actual The question of analogous identities remained expansion to a high power of x, but it has not been open until 1974, when the following result appeared established. As with the first identity, MacMahon [15]. notes the implications for partitions: Theorem 3. For 1 ≤ i ≤ k, states that of any given number such parti- N2+···+N2 +N +N +···+N tions are equi-numerous with those whose X q 1 k−1 i i+1 k−1 parts are all of the forms 5m + 2, 5m + 3. (q)n1 (q)n2 ··· (q)nk−1 n1,...,nk−1≥0 So here in 1916 was an immensely appealing ∞ Y 1 unsolved problem. As Hardy remarks [14, p. 91]: = , n n=1 1 − q Ramanujan rediscovered the formulae some- n6≡0,±i(mod 2k+1) time before 1913. He had then no proof (and where N = n + n + · · · + n , and (q) = knew that he had none), and none of the j j j+1 k−1 n (1 − q)(1 − q2) ··· (1 − qn). mathematicians to whom I communicated the formulae could find one. They are therefore Since the 1970s there has been a cornucopia stated without proof in the second volume of Rogers–Ramanujan-type results pouring forth. of MacMahon’s Combinatory Analysis. As mentioned earlier, Berndt [9, p. 77] gives many The mystery was solved, trebly, in 1917. references to recent work. In that year Ramanujan, looking through It should be added that Schur (mentioned in old volumes of the Proceedings of the Lon- Hardy’s brief history given above) had a strong in- don Mathematical Society, came accidentally fluence on the combinatorial/arithmetical aspects across Rogers’ paper. I can remember very of Rogers-Ramanujan theory. Both his indepen- well his surprise, and the admiration which dent discovery of the Rogers-Ramanujan identities he expressed for Rogers’ work. A correspon- [20] and his modulus 6 theorem of 1926 [21] dence followed in the course of which Rogers greatly impacted subsequent developments. These was led to a considerable simplification of his include the Ph.D. thesis of H. Göllnitz [11] and the original proof. About the same time I. Schur, “weighted words” method of Alladi and Gordon who was then cut off from England by the [2], the latter culminating in a generalization of war, rediscovered the identities again. Göllnitz’s deep modulus 12 theorem by Alladi, In his account of Ramanujan’s notebooks, Berndt Andrews, and Berkovich [3]. [9, pp. 77–79] gives a detailed history of subsequent The analytic side of the study was greatly work on the Rogers-Ramanujan identities. assisted by the discovery of “Bailey chains” in Surprisingly, this aspect of the theory of par- 1984 [6]. Further accounts of this part of the titions languished for the next forty years. W. N. Rogers-Ramanujan identities are provided in S. O. Bailey [8], [7] and his student Lucy Slater [24], [23] Warnaar’s survey [26] “50 Years of Bailey’s lemma” did generalize the series-product identities, but and in Chapter 3 of the Selected Works of George E. D. H. Lehmer [15] and H. L. Alder [1] published Andrews [4]. papers suggesting that there were no generaliza- The Rogers-Ramanujan identities also lead tions of an obvious sort. Even directly to a study of a related , [18, p. 73] asserted that “It can be shown there can a topic dealt with at lenghth in [9].

1530 Notices of the AMS Volume 59, Number 11 References [1] Henry L. Alder, The nonexistence of certain identities in the theory of partitions and compositions, Bull. Amer. Math. Soc., 54:712–722, 1948. [2] Krishnaswami Alladi, The method of weighted words and applications to partitions. In Number The- ory (Paris, 1992–1993), volume 215 of London Math. Soc. Lecture Note Ser., pages 1–36, Cambridge Univ. Press, Cambridge, 1995. [3] Krishnaswami Alladi, George E. Andrews, and Alexander Berkovich, A new four parameter q- series identity and its partition implications, Invent. Math., 153(2):231–260, 2003. [4] G. E. Andrews and A. V. Sills, Selected Works Photo courtesy of Krishnaswami Alladi. of George E. Andrews: With Commentary, Imperial College Press, 2012. The only cot at the humble home of Ramanujan [5] George E. Andrews, An analytic generalization of in Kumbakonam. As a young boy, Ramanujan the Rogers-Ramanujan identities for odd moduli, Proc. used to sit on the windowsill and do his “sums”, Nat. Acad. Sci. U.S.A., 71:4082–4085, 1974. watching the passersby on the street. [6] , Multiple series Rogers-Ramanujan type identities, Pacific J. Math., 114(2):267–283, 1984. [7] W. N. Bailey, Some identities in combinatory analysis, Phys.-Math. Kl., pages 302–321, 1917. Reprinted in Proc. London Math. Soc. (2), 49:421–425, 1947. Gesammelte Abhandlungen. Springer-Verlag, 1973. [8] , Identities of the Rogers-Ramanujan type, Proc. [21] , Zur additiven Zahlentheorie, S.-B. Preuss. Akad. London Math. Soc. (2), 50:1–10, 1948. Wiss. Phys.-Math. Kl., pages 488–495, 1926. Reprinted [9] Bruce C. Berndt, Ramanujan’s Notebooks. Part III, in Gesammelte Abhandlungen, Springer-Verlag, 1973. Springer-Verlag, New York, 1991. [22] A. Selberg, Collected Papers, Vol. I, Springer-Verlag, [10] , Zachary A. Kent, and Ken Ono, 1989. l-adic properties of the partition function, Advances [23] L. J. Slater, A new proof of Rogers’s transformations in Mathematics, 229(3):1586–1609, 2012. of infinite series, Proc. London Math. Soc. (2), 53:460– [11] H. Göllnitz, Partitionen mit Differenzenbedingungen, 475, 1951. J. Reine Angew. Math., 225:154–190, 1967. [24] , Further identities of the Rogers-Ramanujan [12] Basil Gordon, A combinatorial generalization of the type, Proc. London Math. Soc. (2), 54:147–167, 1952. Rogers-Ramanujan identities. Amer. J. Math., 83:393– [25] R. C. Vaughan, The Hardy-Littlewood Method, Cam- 399, 1961. bridge Tracts in Mathematics, vol. 80, Cambridge [13] G. H. Hardy and S. Ramanujan, Asymptotic formulæ University Press, Cambridge, 1981. in combinatory analysis [Proc. London Math. Soc. (2) [26] S. Ole Warnaar, 50 years of Bailey’s lemma, in Alge- 17 (1918), 75–115], in Collected Papers of Srinivasa braic Combinatorics and Applications (Gößweinstein, Ramanujan, pages 276–309, AMS Chelsea Publ., Amer. 1999), pages 333–347, Springer, Berlin, 2001. Math. Soc., Providence, RI, 2000. [14] G. H. Hardy, Ramanujan: Twelve Lectures on Sub- jects Suggested by His Life and Work, AMS Chelsea Bruce C. Berndt Publishing Series, Amer. Math. Soc., Providence, RI, 1999. Ramanujan’s Earlier Notebooks and His Lost [15] D. H. Lehmer, Two nonexistence theorems on partitions, Bull. Amer. Math. Soc., 52:538–544, 1946. Notebook [16] Percy A. MacMahon, Combinatory Analysis. Vols. I, II (bound in one volume), Dover Phoenix Editions, Since the centenary of Ramanujan’s birth, a large Dover Publications Inc., Mineola, NY, 2004. Reprint of amount of Ramanujan’s mathematics that had An Introduction to Combinatory Analysis (1920) and been ensconced in his earlier notebooks and Combinatory Analysis. Vols. I, II (1915, 1916). in his lost notebook has been brought to light. [17] H. Rademacher, On the partition function p(n), Proc. In particular, in 1987 only one book had been London Math. Soc., 43(2):241–254, 1937. [18] , Lectures on Elementary Number Theory, Tata written on Ramanujan’s notebooks [2], but now Institute of Fundamental Research, Bombay. Lec- several have appeared [2], [1]. We now have a tures on mathematics and physics. Mathematics. Tata larger portion of Ramanujan’s mathematics to Institute of Fundamental Research, 1963. appreciate, evaluate, analyze, and challenge. The [19] Srinivasa Ramanujan, Collected Papers of Srinivasa challenge is to ascertain Ramanujan’s thinking and Ramanujan, AMS Chelsea Publishing, Amer. Math. Soc., motivation and to properly place his work within Providence, RI, 2000, edited by G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, third printing of the 1927 contemporary mathematics. Since other writers in original, with a new preface and commentary by Bruce this series of articles are focusing on Ramanujan’s C. Berndt. [20] I. Schur, Ein Beitrag zur additiven Zahlentheorie und Bruce C. Berndt is professor of mathematics at the Univer- zur Theorie der Kettenbrüche, S.-B. Preuss. Akad. Wiss. sity of Illinois, Urbana-Champaign. His email address is [email protected].

December 2012 Notices of the AMS 1531 lost notebook, we give more concentration in this not until March 1925 that Hardy returned the paper on his earlier notebooks. first notebook to the through For most of his life, both in Kumbakonam and its librarian, S. R. Rangathanan [15, pp. 56, 57]. A later in Madras, Ramanujan worked on a slate. handwritten copy was subsequently made and sent Paper was expensive for him, and so he recorded to Hardy in August 1925. only his final results in notebooks. Writing down Meanwhile, Hardy had originally wanted to bring his proofs in notebooks would have taken precious Ramanujan’s published papers, his notebooks, and time; Ramanujan was clearly anxious to get on any other unpublished manuscripts together for with his next ideas. Moreover, if someone had publication. To that end, in a letter to Hardy dated asked him how to prove a certain claim in his August 30, 1923, Francis Dewsbury, registrar at notebooks, he undoubtedly was confident that the University of Madras, wrote, “I have the honour he could reproduce his argument. We do not to advise despatch to-day to your address per know precisely when Ramanujan began to record registered and insured parcel post of the four his mathematical discoveries in notebooks, but manuscript note-books referred to in my letter it was likely around the time that he entered No. 6796 of the 2nd idem. I also forward a packet of the Government College in Kumbakonam in 1904. miscellaneous papers which have not been copied. Except for possibly a few entries, he evidently It is left to you to decide whether any or all of stopped recording his theorems in notebooks after them should find a place in the proposed memorial he boarded a passenger ship for England on March volume. Kindly preserve them for ultimate return 17, 1914. That Ramanujan no longer concentrated to this office.” The “four manuscript notebooks” on logging entries in his notebooks is evident from were handwritten copies of the second and third two letters that he wrote to friends in Madras notebooks, and the packet of miscellaneous papers during his first year in England [5, pp. 112–113; evidently included the “lost notebook”, which 123–125]. In a letter of November 13, 1914, to his Hardy passed on to Watson and which was later friend R. Krishna Rao, Ramanujan confided, “I have rediscovered by George Andrews in the Trinity changed my plan of publishing my results. I am College Library in March 1976. Note that, in not going to publish any of the old results in my contradistinction to Dewsbury’s request, Hardy notebooks till the war is over.” And in a letter of never returned the miscellaneous papers. January 7, 1915, to S. M. Subramanian, Ramanujan As it transpired, only Ramanujan’s published admitted, “I am doing my work very slowly. My papers were collected for publication. However, notebook is sleeping in a corner for these four published with the Collected Papers [12] were or five months. I am publishing only my present the first two letters that Ramanujan had written researches as I have not yet proved the results in to Hardy, which contained approximately one my notebooks rigorously.” hundred twenty mathematical claims. Upon their Prior to his journey to England in 1914, Ra- manujan prepared an enlarged edition of his first publication, these letters generated considerable notebook; this augmented edition is the second interest, with the further publication of several notebook. A third notebook contains only thirty- papers establishing proofs of these claims. Con- three pages. G. N. Watson, in his delightful account sequently, either in 1928 or 1929, at the strong of the notebooks [16], relates that Sir , suggestion of Hardy, Watson and B. M. Wilson, head of the Indian Meteorological Observatory and one of the three editors of [12], agreed to edit the a strong supporter of Ramanujan, told him that in notebooks. The word “edit” should be interpreted April 1912 he observed that Ramanujan possessed in a broader sense than usual. If a proof of a claim four or five notebooks, each with a black cover and can be found in the literature, then it is sufficient each about an inch thick. We surmise that these to simply cite sources where proofs can be found; notebooks were either a forerunner of the first if a proof cannot be found in the literature, then notebook or that the first notebook is an amalgam the task of the editors is to provide one. In an of these earlier black-covered notebooks. address [16] to the London Mathematical Society When Ramanujan returned to India on March 13, on February 5, 1931, Watson cautioned (in retro- 1919, he took with him only his second and spect, far too optimistically), “We anticipate that third notebooks. The first notebook was left with it, together with the kindred task of investigating G. H. Hardy, who used it to write a paper [9] on a the work of other writers to ascertain which of his chapter in the notebook devoted to hypergeometric results had been discovered previously, may take series. Hardy’s paper [9] was read on February 5, us five years.” Wilson died prematurely in 1935, 1923, and published later during that year. He and although Watson wrote approximately thirty begins by writing, “I have in my possession a papers on Ramanujan’s work, his interest evidently manuscript note-book which Ramanujan left with flagged in the late 1930s, and so the editing was me when he returned to India in 1919.” It was not completed.

1532 Notices of the AMS Volume 59, Number 11 ihtefis otiigtefis oeokand notebook first the containing first the with 9712,Rmnjnscnetaino number on concentration Ramanujan’s 1917–1920, unorganized the in material considerable and 16–21 December volumes, two in set were notebooks The taken. Institute [ a Tata edition published photocopy Bombay the in when Research Fundamental public of the to available through was theory and 1912–1914 years the theta in on Especially particular, functions. in functions, elliptic of distinctive theory Ramanujan’s on focus follow that pages Chapters functions. theta to devotion to enormous an begins he chapter this examine in a since represents notebook point, second turning the in forms. 16 modular Chapter with and connections combinatorics having some enumerative also with discoveries and his theory of being number analysis in with work associated his of much with elliptic and functions, by theory, number analysis, examined under of ten purview primarily the topics fall to notebooks five his The results in than the Ramanujan incorrect. more of are no all claims perhaps Almost correct; replaced fraction. are be larger should a them, likely wrote by he more that two-thirds time the and at Ramanujan’s new of were two-thirds claims least on at high; contrary, too the much is estimate were This results rediscoveries. these of two-thirds Hardy over that proof. surmised without all almost claims, thousand Sannidhi Street. of Sarangapani front on the home from Ramanujan’s seen the as of Temple tower) Sarangapani (entrance gopuram majestic The loehr h oeok oti vrthree over contain notebooks the Altogether, ial,i 97 h oeok eemade were notebooks the 1957, in Finally, Photo courtesy of Krishnaswami Alladi. q sre o h rttm n lot begin to also and time first the for -series 2012 13 q sre n litcfunctions. elliptic and -series ,btn dtn a under- was editing no but ], oie fteAMS the of Notices isnseriratmta dtn h notebooks the editing at attempt earlier Wilson’s ihtescn opiigtescn n third and second the comprising second the with rt bu aaua’ icvre nhslost his in discoveries Ramanujan’s about write nrw,KnOo n l ana,i particular, in Warnaar, Ole and Ono, Ken Andrews, h epo oyo h oe rmWto and Watson from notes the of copy proving a With to of notebooks. help the attention the in his claims Ramanujan’s of of author all all the devote 1977 May to In began prove. not our assertions could further using we thousand that proved few a two be in but could formulas proved methods, more that few had notebooks a we the class found We that general earlier. using series years a by Eisenstein for formulas of formula these transformation prove observed a could we proved, we were that notebooks the from [ Grosswald Emil Ramanujan’s birth. of anniversary 125th the com- to memorate 2011 in published was new, edition a clearer, much technology, improved photographed. vastly work by Buttressed scratch Ramanujan’s with faithful, even quite is reproduction The notebooks. aucit ilb xmndi h orhbook fourth the in examined Mittag- be will incomplete the manuscripts and fraction of fragments partial The knowledge developing decompositions. firm for Ramanujan Theorem a hand, Leffler have at cases not the did in correctly. particular, always In not but Ramanujan that employed subject occasionally a analysis, complex the of on two use depends least material at discarded In the papers. instances, to his not in chose results Ramanujan these why include ask to natural then [ is example, for papers, portions published for of intended originally were that material 1978 until literature the [ in established not result a diophantine best to approximation the derives Ramanujan these, of one In approximation. diophantine on manuscripts [ Second, problems. these attacking at aimed clearly notebook strong a had problem Ramanujan the in First, interest series. two [ the series with two papers, in Zaharescu of with Alexandru examined and first have Kim we Sun We insight that ideas. topics valuable two and mention provide interests that Ramanujan’s into topics other manuscripts on of fragments and manuscripts [ partial notebook lost the with published by dominated is which notebook, [ volumes five years. in twenty over completed slightly was mathematicians, task other several the of help the with and 6 .As pern n[ in appearing Also ]. nFbur 94 hl edn w aesby papers two reading while 1974, February In nti eiso rilsfrthe for articles of series this In ihtode omlsi h lost the in formulas deep two with , 14 otistrevr og,partial rough, very three contains ] iceproblem circle e 7 3 2 ,[ ], /a n [ and ] where , 8 nwihsm formulas some which in ] 14 4 en xmlsof examples being ] a r rget of fragments are ] and sanneointeger, nonzero a is q sre.However, -series. iihe divisor Dirichlet Notices 14 10 r several are ] n [ and ] George , 11 2 in ] .It ]. 1533 that the author is writing with Andrews on the lost [14] , The Lost Notebook and Other Unpublished notebook [1]. Papers, Narosa, New Delhi, 1988. We conclude with a few remarks about Ra- [15] S. R. Ranganathan, Ramanujan: The Man and the Mathematician, Asia Publishing House, Bombay, 1967. manujan’s methods. It has been suggested that [16] G. N. Watson, Ramanujan’s notebooks, J. London he discovered his results by “intuition” or by Math. Soc. 6 (1931), 137–153. making deductions from numerical calculations or by inspiration from Goddess Namagiri. Indeed, Jonathan M. Borwein like most mathematicians, Ramanujan evidently made extensive calculations that provided guid- Ramanujan and ance. However, Hardy and this writer firmly believe Since Ramanujan’s 1987 centennial, much new that Ramanujan created mathematics as any other mathematics has been stimulated by uncanny mathematician would and that his thinking can formulas in Ramanujan’s Notebooks (lost and be explained like that of other mathematicians. found). In illustration, I mention the exposition However, because Ramanujan did not leave us any by Moll and his colleagues [1] which illustrates proofs for the vast number of results found in various neat applications of Ramanujan’s Master his earlier notebooks and in his lost notebook, Theorem, which extrapolates the Taylor coefficients we often do not know Ramanujan’s reasoning. As of a function, and relates them to methods of Ramanujan himself was aware, some of his argu- integration used in particle physics. I also note ments were not rigorous by then-contemporary lovely work on the modular functions behind standards. Nonetheless, despite his lack of rigor at Apéry and Domb numbers by Chan and others [6], times, Ramanujan doubtless thought and devised and finally I mention my own work with Crandall proofs as would any other mathematician. on Ramanujan’s arithmetic-geometric continued fraction [12]. References For reasons of space, I now discuss only work [1] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost related directly to pi, and so continue a story started Notebook, Parts I–V, Springer, New York, 2005, 2009, in [9], [11]. Truly novel series for 1/π, based on 2012, to appear. elliptic integrals, were found by Ramanujan around [2] B. C. Berndt, Ramanujan’s Notebooks, Parts I–V, Springer-Verlag, New York, 1985, 1989, 1991, 1994, 1910 [19], [5], [7], [21]. One is √ ∞ 1998. 1 2 2 X (4k)! (1103 + 26390k) [3] B. C. Berndt, S. Kim, and A. Zaharescu, Circle and (1) = . π 9801 (k!)43964k divisor problems, and double series of Bessel functions, k=0 Adv. Math, to appear. Each term of (1) adds eight correct digits. Though [4] , Diophantine approximation of the exponential then unproven, Gosper used (1) for the computation function, and a conjecture of Sondow, submitted for of a then-record 17 million digits of π in 1985, publication. [5] B. C. Berndt and R. A. Rankin, Ramanujan: Letters thereby completing the first proof of (1) [7, Ch. 3]. and Commentary, American Mathematical Society, Soon after, David and Gregory Chudnovsky found Providence, RI, 1995; London Mathematical Society, the following variant,√ which relies on the√ quadratic London, 1995. number field Q( −163) rather than Q( 58), as is [6] C. S. Davis, Rational approximation to e, J. Aus- implicit in (1): tral. Math. Soc. 25 (1978), 497–502. ∞ 1 X (−1)k (6k)! (13591409 + 545140134k) [7] E. Grosswald, Die Werte der Riemannschen (2) = 12 . π (3k)! (k!)3 6403203k+3/2 Zetafunktion an ungeraden Argumentstellen, k=0 Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970 Each term of (2) adds fourteen correct digits. (Were (1970), 9–13. [8] , Comments on some formulae of Ramanujan, a larger imaginary quadratic field to exist with Acta Arith. 21 (1972), 25–34. class number one, there would be an even more [9] G. H. Hardy, A chapter from Ramanujan’s note-book, extravagant rational series for some surd divided Proc. Cambridge Philos. Soc. 21 (1923), 492–503. by π [10].) The brothers used this formula several [10] S. Ramanujan, Some formulae in the analytic theory times, culminating in a 1994 calculation of π to of numbers, Mess. Math. 45 (1916), 81–84. over four billion decimal digits. Their remarkable [11] , On certain trigonometric sums and their appli- story was told in a prize-winning New Yorker article cations in the theory of numbers, Trans. Cambridge Philos. Soc. 22 (1918), 259–276. [18]. Remarkably, (2) was used again in 2010 and [12] , Collected Papers, Cambridge University Press, 2011 for the current record computations of π to Cambridge, 1927; reprinted by Chelsea, New York, five and ten trillion decimal digits respectively. 1962; reprinted by the American Mathematical Society, Providence, RI, 2000. Jonathan M. Borwein is Laureate Professor in the School [13] , Notebooks (2 volumes), Tata Institute of Funda- of Mathematical and Physical Sciences at the Univer- mental Research, Bombay, 1957; second edition, Tata sity of Newcastle (NSW), Australia. His email address is Institute of Fundamental Research, , 2011. [email protected].

1534 Notices of the AMS Volume 59, Number 11 Quartic Algorithm for π 12 The record for computation of π has gone from 10 29.37 million decimal digits in 1986 to ten trillion digits in 2011. Since the algorithm below, which found its inspiration in Ramanujan’s 1914 paper, 10 10 was used as part of computations both then and as late as 2009, it is interesting to compare√ the performance√ in each case: Set a0 := 6 − 4 2 and y := 2 − 1; then iterate 8 0 10 4 1/4 1 − (1 − yk ) y + = , k 1 4 1/4 1 + (1 − yk ) (3) 4 2k+3 2 1975 1980 1985 1990 1995 2000 2005 2010 ak+1 =ak(1 + yk+1) −2 yk+1(1+yk+1 +yk+1). Figure 1. Plot of π calculations in digits (dots) Then ak converges quartically to 1/π; each iteration quadruples the number of correct digits. Twenty- compared with the long-term slope of Moore’s one iterations produce an algebraic number that Law (line). coincides with π to well over six trillion places. This scheme and the 1976 Salamin–Brent scheme [7, Ch. 3] have been employed fre- at http://www.numberworld.org/misc_ quently over the past quarter century. Here runs/pi-5t/announce_en.html. See is a highly abbreviated chronology (based on also [4], in which analysis showing these http://en.wikipedia.org/wiki/Chronology_ digits appear to be “very normal” is made. of_computation_of_pi): Daniel Shanks, who in 1961 computed π to over 100,000 digits, once told Phil Davis that • 1986: David Bailey used (3) to compute a billion-digit computation would be “forever 29.4 million digits of π. This required 28 impossible”. But both Kanada and the Chudnovskys hours on one CPU of the new Cray-2 at NASA achieved that in 1989. Similarly, the intuitionists Ames Research Center. Confirmation using Brouwer and Heyting asserted the “impossibility” the Salamin-Brent scheme took another of ever knowing whether the sequence 0123456789 40 hours. This computation uncovered appears in the decimal expansion of π; yet it was hardware and software errors on the Cray-2. found in 1997 by Kanada, beginning at position • January 2009: Takahashi used (3) to com- 17387594880. As late as 1989, Roger Penrose pute 1.649 trillion digits (nearly 60,000 ventured, in the first edition of his book The times the 1986 computation), requiring Emperor’s New Mind, that we likely will never 73.5 hours on 1,024 cores (and 6.348 Tbyte know if a string of ten consecutive 7s occurs memory) of the Appro Xtreme-X3 system. in the decimal expansion of π. This string was Confirmation via the Salamin-Brent scheme found in 1997 by Kanada, beginning at position took 64.2 hours and 6.732 Tbyte of main 22869046249. memory. Figure 6 shows the progress of π calculations • April 2009: Takahashi computed 2.576 since 1970, superimposed with a line that charts trillion digits. the long-term trend of Moore’s Law. It is worth • December 2009: Bellard computed nearly noting that whereas progress in computing π 2.7 trillion decimal digits (first in binary) exceeded Moore’s Law in the 1990s, it has lagged a using (2). This took 131 days, but he used bit in the past decade. Most of this progress is still only a single four-core workstation with in mathematical debt to Ramanujan. lots of disk storage and even more human As noted, one billion decimal digits were first intelligence! computed in 1989, and the ten (actually fifty) billion • August 2010: Kondo and Yee computed digit mark was first passed in 1997. Fifteen years 5 trillion decimal digits, again using later one can explore, in real time, multibillion step (2). This was done in binary, walks on the hex digits of π at http://carmaweb. then converted to decimal. The binary newcastle.edu.au/piwalk.shtml, as drawn by digits were confirmed by computing 32 Fran Aragon. hexadecimal digits of π ending with 2 position 4,152,410,118,610 using BBP-type Formulas for 1/π and More formulas for π due to Bellard and Plouffe About ten years ago Jésus Guillera found various [7, Chapter 3]. Additional details are given Ramanujan-like identities for 1/π N using integer

December 2012 Notices of the AMS 1535 relation methods. The three most basic—and Calabi-Yau and Supercongruences entirely rational—identities are Motivated by the theory of Calabi-Yau differential (4) equations [2], Almkvist and Guillera have discov- ∞ 4 X  1  ered many new identities. One of the most pleasing = (−1)nr(n)5(13+180n+820n2) 2n+1, π 2 32 is n=0 (5) ∞ 2  ∞ 1 ? 32 X (6 n)! 532 n + 126 n + 9 2 X 1 (11) = . = (−1)nr(n)5(1 + 8n + 20n2) 2n+1, π 2 3 (n!)6 106n+3 π 2 2 n=0 n=0 This is yet one more case where mysterious (6) connections have been found between disparate ∞  2n+1 4 ? X 1 = r(n)7(1+14n+76n2 +168n3) , parts of mathematics and Ramanujan’s work [21], π 3 8 n=0 [13], [14]. where r(n) := (1/2 · 3/2 ····· (2n − 1)/2)/n!. As a final example, we mention the existence of Guillera proved (4) and (5) in tandem by very supercongruences of the type described in [3], [16], ingeniously using the Wilf-Zeilberger algorithm [23]. These are based on the empirical observation N [20], [17] for formally proving hypergeometric-like that a Ramanujan series for 1/π , if truncated identities [7], [15], [21]. No other proof is known. after p − 1 terms for a prime p, seems always to The third, (6), is almost certainly true. Guillera produce congruences to a higher power of p. The ascribes (6) to Gourevich, who found it using formulas below are taken from [22]: integer relation methods in 2001. (12) There are other sporadic and unexplained exam- p−1 1 1 3 3 X ( )n( )n( )n   ples based on other symbols, most impressively a 4 2 4 3 + 34n + 120n2 ≡ 3p2(mod p5), 4n 5 2010 discovery by Cullen: n=0 2 (1)n (7) 11 ∞ 1 1 7 3 2 ? X ( )n( ) ( )n = 4 2 n 4 (13) 4 9 π (1) p−1 1 1 7 3 n=0 n X ( )n( )n( )n   4 2 4 21 + 466n + 4340n2 + 20632n3 + 43680n4 12n 12n 9  1  2 (1)n ×(21 + 466n + 4340n2 + 20632n3 + 43680n4) . n=0 ? 2 ≡ 21p4(mod p9). We shall revisit this formula below. We note that (13) is the supercongruence corre- sponding to (6), while for (12) the corresponding Formulae for π 2 infinite series sums to 32/π 4. We conclude by In 2008 Guillera [15] produced another lovely, if reminding the reader that all identities marked numerically inefficient, pair of third-millennium ? with ‘=’ are assuredly true but remain to be proved. identities—discovered with integer relation meth- Ramanujan might well be pleased. ods and proved with creative telescoping—this time for π 2 rather than its reciprocal. They are based on: References [1] T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Har-  3  2 rison, V. Moll, and A. Straub, Ramanujan’s Master ∞ + 1 ∞ 1 X 1 x 2 X 2 Theorem, Ramanujan J. 29 (2012, to appear). (8) n (6(n + x) + 1) = 8x n , 22n 3 2 [2] G. Almkvist, Ramanujan-like formulas for 1/π 2 à la n=0 (x + 1)n n=0 (x + 1)n Guillera and Zudilin and Calabi-Yau differential equa- and tions, Computer Science Journal of Moldova 17, no. 1 (49) (2009), 100–120. (9) [3] G. Almkvist and A. Meurman, Jesus Guillera’s for-  3  2 ∞ x + 1 ∞ x + 1 mula for 1/π 2 and supercongruences (Swedish), X 1 2 n X 2 n (42(n + x) + 5) = 32x . Normat 58, no. 2 (2010), 49–62. 26n (x + 1)3 (2x + 1)2 n=0 n n=0 n [4] D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Here (a)n = a(a + 1) ··· (a + n − 1) is the rising Dinneen, M. Dumitrescu, and A. Yee, An empir- . Substituting x = 1/2 in (8) and (9), he ical approach to the normality of pi, Experimental obtained respectively the formulae Mathematics. Accepted February 2012. [5] N. D. Baruah, B. C. Berndt, and H. H. Chan, Ramanu- ∞ jan’s series for 1/π: A survey, Amer. Math. Monthly X 1 (1)3 π 2 (10) n (3n + 2) = , 116 (2009), 567–587. 22n  3 3 4 n=0 [6] B. C. Berndt, H. H. Chan, and S. S. Huang, Incomplete 2 n ∞ elliptic integrals in Ramanujan’s lost notebook, pp. X 1 (1)3 π 2 n (21n + 13) = 4 . 79–126 in q-series from a Contemporary Perspective, 26n  3 3 3 n=0 Contemp. Math., 254, Amer. Math. Soc., Providence, 2 n RI, 2000.

1536 Notices of the AMS Volume 59, Number 11 [7] J. M. Borwein and D. H. Bailey, Mathematics by Ex- periment: Plausible Reasoning in the 21st Century, 2nd ed., A K Peters, Natick, MA, 2008. [8] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Ex- perimentation in Mathematics: Computational Roads to Discovery, A K Peters, 2004. [9] J. M. Borwein and P. B. Borwein, Ramanujan and pi, Scientific American, February 1988, 112–117. Reprinted in Ramanujan: Essays and Surveys, B. C. Berndt and R. A. Rankin, eds., AMS-LMS , vol. 22, 2001, pp. 187–199. [10] , Class number three Ramanujan type series for 1/π, Journal of Computational and Applied Math. (Special Issue) 46 (1993), 281–290. [11] J. M. Borwein, P. B. Borwein, and D. A. Bailey, Ra- DID YOU manujan, modular equations and pi or how to compute a billion digits of pi, MAA Monthly 96 (1989), 201–219. Reprinted on http://www.cecm.sfu.ca/organics, 1996. [12] J. M. Borwein and R. E. Crandall, On the Ramanujan KNOW AGM fraction, Part II: The complex-parameter case, that Elsevier: Experimental Mathematics 13 (2004), 287–296. [13] H. H. Chan, J. Wan, and W. Zudilin, Legendre poly- nomials and Ramanujan-type series for 1/π, Israel J. Math., in press, 2012. [14] , Complex series for 1/π, Ramanujan J., in press, 2012. [15] J. Guillera, Hypergeometric identities for 10 ex- tended Ramanujan-type series, Ramanujan J. 15 (2008), 219–234. Has opened access to over 155,000 [16] J. Guillera and W. Zudilin, “Divergent” Ramanujan- archived mathematics articles type supercongruences, Proc. Amer. Math. Soc. 140 (2012), 765–777. [17] M. Petkovsek, H. S. Wilf, and D. Zeilberger, A = B, Is continuing to lower A K Peters, 1996. its journal list prices [18] R. Preston, The Mountains of Pi, New York- er, March 2, 1992, http://www.newyorker.com/ Continues to support and archive/content/articles/050411fr_archive01. expand initiatives for the [19] S. Ramanujan, Modular equations and approx- mathematics community imations to pi, Quart. J. Math. 45 (1913–14), 350–372. Is a founding partner of [20] H. S. Wilf and D. Zeilberger, Rational functions cer- Research4Life, providing tify combinatorial identities, Journal of the AMS 3 developing countries with access (1990), 147–158. [21] W. Zudilin, Ramanujan-type formulae for 1/π: a sec- ond wind? Modular forms and string duality, in Fields Inst. Commun., 5, Amer. Math. Soc., Providence, RI, 2008, pp. 179–188. [22] , Arithmetic hypergeometric series, Russian Math. Surveys 66:2 (2011), 1–51. [23] , Ramanujan-type supercongruences, J. Number Theory 129 (2009), 1848–1857. The second installment of this article—with pieces FOR MORE by Ken Ono, K. Soundararajan, R. C. Vaughan, and S. Ole Warnaar—will appear in the January 2013 INFORMATION : issue of the Notices. www.elsevier.com/mathematics

Mathematics

December 2012 Notices of the AMS 1537