Other Titles in This Series

Volume 9 Bruc e C. Berndt and Robert A. Rankin Ramanujan: Letter s and commentar y 1995 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993

5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 M/p

SRINIVASA RAMANUJA N 1887-1920 https://doi.org/10.1090/hmath/009

History of Mathematics Volume 9

RAMANUJAN LETTERS and COMMENTAR Y

BRUCE C. BERNDT and ROBER T A. RANKI N

American Mathematical Society London Mathematical Society 2000 Mathematics Subject Classification. Primar y 01A32 , 01A60 , 01A70 ; Secondary 11-03 , 40-03 , 41-03 .

Photograph o n fron t cover : Indi a postag e stam p honorin g Ramanujan . Bac k cover : Ramanujan wit h friend s a t Trinit y College . Backgroun d photo : Lette r from Ramanuja n to hi s father . A lis t o f photograph credit s i s include d a t th e beginnin g o f this volume .

Library o f Congres s Cataloging-in-Publicatio n Dat a Ramanujan Aiyangar , Srinivasa , 1887-1920 . [Correspondence, Selections ] Ramanujan : letter s and commentar y / Bruc e C . Berndt, Rober t A . Rankin . p. cm . — (Histor y o f mathematics, ISS N 0899-242 8 ; v. 9 ) Includes bibliographica l reference s (p . - ) and index . ISBN 0-8218-0287- 9 (hardcover ) (acid-free ) ISBN 0-8218-0470- 7 (softcover ) (acid-free ) 1. Ramanujan Aiyangar , Srinivasa, 1887-1920—Correspondence . 2 . Mathematicians—India — Correspondence. 3 . Number theory. I . Berndt, Bruc e C, 1939 - . II . Rankin, Rober t A . (Rober t Alexander), 1915 - . III . Title. IV . Series . QA29.R3A4 199 5 510'.92—dc20 95-525 4

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s granted t o quot e brie f passage s from thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the source i s given. Republication, systematic copying, or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e from th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o reprint-permissionQams.org .

© Copyrigh t 199 5 by the American Mathematica l Society . Al l right s reserved . Reprinted wit h correction s b y the America n Mathematica l Society , 1997 . Printed i n the Unite d State s o f America . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . @ Th e pape r use d i n this boo k i s acid-free an d fall s within the guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e page a t http://www.ams.org / The Londo n Mathematica l Societ y i s incorporated unde r Roya l Charte r and i s registered wit h th e Charit y Commissioners . 10 9 8 7 6 5 4 3 2 1 2 1 1 1 0 09 08 0 7 To our wives, Helen and Mary This page intentionally left blank Contents

Preface i x

Photograph Credits xii i

A Brief Biography of Ramanujan 1

Chapter 1. Ramanujan in Madras 7

Chapter 2. Ramanujan's First Two Letters to Hardy and Hardy's

Response 2 1

Chapter 3. Preparing to Go to England 7 3

Chapter 4. Ramanujan at Cambridge 10 9

Chapter 5. Ramanujan is 11 1 14 5

Chapter 6. Ramanuja n Returns to India 20 3

Chapter 7. After his Death 22 7

Chapter 8. Ramanujan's Papers an d Manuscripts 25 9

Chapter 9. Family Histories 31 3

Provenance of the Letters 31 7

References 32 3

Index 33 5 This page intentionally left blank Preface

Ramanujan's lette r of 1 6 January 191 3 to G.H. Hardy is one of the most famous letter s in the history o f mathematics. Afte r introducin g himsel f "a s a cler k i n th e Account s Departmen t o f th e Por t Trus t Offic e a t Madras, " Ramanujan bega n t o relat e som e o f hi s mathematica l discoveries . I n thi s and the followin g letter of 2 7 February 1913 , Ramanujan se t forth ove r one hundred o f hi s theorems. Th e da y o n whic h Hard y receive d Ramanujan' s first lette r mark s th e dawnin g o f th e recognitio n o f Ramanujan' s remark - able mathematical talent s outsid e hi s nativ e sout h India . Later , i n th e lat e 1920s, after Ramanujan's death , these letters spawned several papers by G.N. Watson, C.T. Preece , an d others . Indeed , th e influenc e o f thes e letters stil l pervades contemporary researc h i n mathematics. Correspondence between Ramanujan an d Hardy occurred not only befor e Ramanujan journeye d t o Englan d i n Marc h 1914 , bu t als o fo r tw o year s while Ramanujan wa s confined t o various nursing homes in 1917-1919 . Af - ter returnin g to India , Ramanuja n wrot e to Hardy onl y once . Thes e letter s contain muc h interestin g mathematic s a s wel l a s update s o n Ramanujan' s illness. Unfortunately , som e of the correspondence between thes e two great mathematicians has been lost. Of course, Ramanujan wrot e many letters to his family an d friend s fro m Cambridge. Althoug h most of these letters have been lost, several have been preserved. W e ow e a great deb t t o P.K . Srinivasan , wh o collecte d man y o f these letter s an d mad e the m availabl e t o th e genera l publi c [263] . Othe r letters ar e secure d i n Cambridg e Universit y Librar y an d i n th e librar y o f Trinity College , Cambridge ; photocopie s o f som e o f thes e ar e publishe d i n Ramanujan's "los t notebook " [227 ] an d elsewhere . Furthe r letters , mostl y from the files of the Madras Port Trust Office, are safeguarded in the National Archives in Delhi. The presen t author s hav e collecte d a s man y letter s to , from , an d abou t Ramanujan a s ha s bee n possible . A list o f th e letters , togethe r wit h thei r sources, is given in an appendix. Ther e are several motivations for collectin g these letters and writing extensive commentary o n them. First, a s indicate d above , th e correspondenc e betwee n Ramanuja n an d

IX X PREFACE Hardy contains much fascinating mathematics. W e have examined each the- orem quoted b y Ramanujan in these letters an d have trace d its history an d influence. I t is hoped that this scholarship will be helpful t o mathematicians who have been influenced by Ramanujan's mathematics. Second, man y of th e people wit h whom Ramanuja n corresponde d o r are mentioned i n thi s correspondenc e ar e known t o mos t o f u s toda y onl y b y name. Thus , shor t biographie s o f mos t o f th e correspondent s hav e bee n prepared. Fo r example , whil e searchin g fo r informatio n abou t Si r Gilber t Walker, Hea d of th e Meteorological Observator y i n Madra s and one of th e first to recogniz e Ramanujan' s mathematica l gifts , w e discovered tha t h e is well known to meteorologists today as one of the founders of modern weather forecasting. Third, man y part s of Ramanujan' s persona l letter s nee d to be explained, especially to readers in the West unfamiliar with India n places, culture, and foods. Fourth, P.K. Srinivasan's collection [263] has been out of print for several years, and very few copies ar e available outside India. Thus , we are anxious that these letter s and others not i n Srinivasan' s boo k be mad e accessible t o those both within and outside India who have an interest in Ramanujan and his mathematics. It should be emphasized that this work is not meant to be a full biography of Ramanujan. Onl y those parts of Ramanujan's life an d work arising from the letters that we have collected ar e discussed. Thos e desiring a more com- plete an d detaile d accoun t o f Ramanujan' s lif e ar e advised t o rea d Rober t Kanigel's boo k [142] . Les s complet e biographie s hav e bee n writte n b y S . Ram [195 ] and S.R . Ranganatha n [229] . Prio r t o Kanigel' s book , th e mos t commonly rea d accoun t o f Ramanujan' s lif e wa s tha t o f P.V . Sesh u Aiya r and R. Ramachandra Rao, found a t the beginning of Ramanujan's Collected Papers [226]. Thi s is an amalgam of earlie r obituaries written by them indi- vidually [1J , [243]. Hardy' s obituary [98], [112, pp. 702-720], [226, pp. xxix- xxvi] focuses mor e on Ramanujan's mathematical contributions . Almos t all of th e man y brie f biographie s o f Ramanuja n tha t hav e bee n writte n sinc e Ramanujan's deat h ar e based on th e article s of Sesh u Aiyar , Ramachandr a Rao, and Hardy. Bernd t [36] has written a brief account of Ramanujan's life and mathematics fo r nonspecialists . Anecdote s an d persona l reminiscence s about Ramanujan can be found i n the collections of Srinivasa n [263 ] and R. Bharathi [45]. In additio n t o th e letter s printe d here , w e als o includ e som e newspape r articles published during Ramanujan's lifetime, som e office memo s from the Madras Por t Trus t Office , an d jottings fro m a family diary . I n most cases, we follow each item with commentary. The authors are extremely grateful t o many people who helped secure let- ters an d othe r materia l fo r thi s volum e an d wh o helpe d u s i n writin g ou r commentaries. PREFACE XI

The conception for this work began shortly after S. Chandrasekhar, through the cooperation of th e lat e Prim e Ministe r Raji v Gandhi , obtaine d fo r on e of u s (BCB ) copie s o f al l th e Ramanuja n correspondenc e i n th e Nationa l Archives in Delhi. Sinc e that time, Chandrasekha r has offered considerabl e valuable advice and obtained further information fo r us. We again expres s our gratitude t o P.K. Srinivasa n wh o long ag o realized that letter s o f Ramanuja n woul d b e los t unles s h e mad e effort s t o collec t them. K. Srinivasa Rao has aided us in several ways. Throug h his efforts, we ob- tained copies of the letters of Ramanujan to his boyhood friend, E . Vinayaka Row. Srinivas a Ra o ha s provide d considerabl e informatio n abou t names , places, an d customs in India an d has offere d man y useful suggestions , afte r a very careful readin g of our manuscript. We are thankful tha t Kanigel [142 ] has brought Ramanujan's story to the general public, and information tha t he gathered has been useful t o us. We are also pleased to thank George Andrews for providing commentar y on Ramanujan' s las t lette r t o Hardy ; Richar d Aske y fo r man y helpfu l re - marks; Krishnaswam i Allad i an d S . Bhargav a fo r informatio n abou t Indi a and its customs; Paul Bialek for comments on Ramanujan's letters to Hardy written from nursing homes; John Brillhart for writing the biography of D.H. Lehmer; Dan Grayso n fo r answerin g hundred s of querie s abou t TjX ; V.S . Ramachandran, Curato r of Visvesvaray a Industria l an d Technologica l Mu - seum, Bangalore, for several of the photographs appearing in this volume; A. Ranganathan for information abou t his grandfather, Dr . P.S. Chandrasekar, who tende d Ramanuja n i n th e fina l yea r o f hi s life ; T.V . Rangaswam i fo r a copy of Ramanujan' s passport ; Ranga an d Shantha Rao fo r muc h knowl- edge abou t India n foods an d culture; C.A. Redd i fo r informatio n abou t his father-in-law, A.S . Ramalingam, who was a close friend of Ramanujan, and for obtaining fo r u s documents fro m th e Tamil Nad u Archives ; K . Venkat - achaliengar fo r severa l helpfu l comments ; V . Viswanatha n fo r informatio n about his grandfather, S . Narayana Aiyar ; K.C. Wal i fo r helpful correspon - dence; an d Wayn e Wendlan d fo r informatio n o n Si r Gilber t Walke r an d weather forecasting. Many people in several organizations have kindly helped us in uncovering information about those whose names appear in the correspondence detailed in this book. Thus , w e offe r ou r sincere thank s to librarian s a t the Univer- sity o f Illinois , i n particular , Nanc y Anderso n an d Mar y Berney ; Glasgo w University Library ; Mitchell Library , Glasgow ; Trinity College , Cambridge , in particular , th e Maste r an d Fellow s o f Trinity , th e Librarians , th e Col - lege Office , an d Rosemary Graham , the Manuscrip t Cataloguer ; the Univer- sity Archives at the Cambridge University Library ; the Cambridge Scientifi c Periodicals Library ; th e Cambridg e Philosophica l Society ; th e Biographica l Assistant an d Archivis t a t St . John' s College , Cambridge ; th e Archivis t a t Merton College, Oxford ; the Archivist a t Christ Church , Oxford; J . Vernon Xll PREFACE Armitage, College of St. Hild and St. Bede, Durham; Pacelli Clanchy, Valua- tion Office, Dublin; David H. Fowler, University of Warwick; Donald Glynn, Rutland Centre, Dublin; Wilbur R. Knorr, Stanford University ; Walter Led- ermann, University of Sussex ; the late William McLean, formerly Registra r of the University o f Madras; Peter Neumann, the Queen's College, Oxford ; David J. Simms , Trinity College , Dublin; Wolfgan g Schwarz , University of Frankfurt a m Main; the late Gertrude K. Stanley, formerly o f Westfield Col- lege, London; M.G. Underwood, Archivist at St. John's College, Cambridge; the Librarian of the Royal Society; the Administrator o f the London Mathe- matical Society; the India Office Library; and the Office Manage r at Barclays Bank. We note, in conclusion, that the copyright of all Ramanujan's writings , of every kind, was, after his death, purchased from his widow by the University of Madras. This project could not have been begun if it were not for a grant from the Alfred P. Sloan Foundation, to which we offer ou r sincere appreciation. Photograph Credit s

The American Mathematical Societ y gratefully acknowledge s the kind- ness of thes e individual s an d institutions i n grantin g th e followin g permis - sions:

V. S . Ramachandran , Curator , Visvesvaray a Industria l an d Technologica l Museum, Bangalore, India Reproduction o f th e cover, G . S . Carr , Synopsis of Elementary Results Pure Mathematics, Franci s Hodgson , London , 1880 , p . 5 ; courtesy o f V. S . Ramachandra n Photo o f Pachaiyappa' s College , Madras , p . 6 ; courtes y o f V . S . Ra - machandran Photo of R . Ramachandra Rao, p. 6; courtesy of V . S . Ramachandra n Reproduction of Ramanujan's letter to Madras Port Trust, p. 9; courtesy of V. S . Ramachandran Photo of th e old Por t Trus t Offic e a t Madras , p. 10 ; courtesy o f V . S . Ramachandran Photo of S . Narayana Aiyar, p. 74; courtesy of V . S . Ramachandra n Photo of E . H. Neville, p . 95; courtesy of V . S . Ramachandran Photo of Ramanuja n with friends a t Trinity College, p. 11 5 and cover; courtesy of V. S . Ramachandran Reproduction o f Ramanujan' s lette r to hi s father , p . 120 ; courtesy o f V. S. Ramachandra n Photo of Ramanujan with western clothing and haircut, p. 127; courtesy of V. S. Ramachandran Photo o f Ramanuja n an d fello w graduates , p . 137 ; courtes y o f V . S . Ramachandran Photo of A . S . Ramalingam, p . 163 ; courtesy of V . S . Ramachandra n

Xlll xiv PHOTOGRAP H CREDIT S Photo of Gometra , Madras, p. 231; courtesy of V. S . Ramachandran Photo of Komalatammal, p . 236; courtesy of V. S . Ramachandran Photo of Janak i Ammal, p. 239; courtesy of V . S . Ramachandra n Reproduction o f postag e stamp , p . 31 4 an d cover ; courtes y o f V . S . Ramachandran Oxford University Pres s Photo of G. H. Hardy, p. 31; from G. H. Hardy, Collected Papers, Vol. 1, Clarendon Press, 1966 ; by permission of Oxford University Pres s University of Dunde e Photo o f B . M . Wilson , p . 268 ; copyrigh t Universit y o f Dundee ; b y permission of University of Dunde e The Royal Societ y Photo o f G . N . Watson , p . 282 ; by permissio n o f Th e Roya l Society , London S. Chandrasekha r Photo of S. Ramanujan, frontispiece; by permission of S. Chandrasekhar Town High School, Kumbakonam, Indi a Reproduction of certificate awarde d to Ramanujan, p. 4 All other photos were provided by the authors. References

1. P. V. Seshu Aiyar, "The late Mr. S. Ramanujan, B.A. , F.R.S.," /. Indian Math. Soc. 12(1920), 81-86. 2. S . N. Aiyar, "The distribution of primes," /. Indian Math. Soc. 5(1913), 60-61. 3. K . Ananda Rau , "O n the convergence an d summabilit y o f Dirichlet' s series, " Proc. London Math. Soc. (2) 34 (1932), 414-440. 4. G . E. Andrews, "On the theorems of Watson and Dragonette for Ramanujan' s mock theta functions, " Amer. J. Math. 8 8 (1966), 454-490. 5. G . E . Andrews, "O n ^-differenc e equation s fo r certai n well-poise d basi c hy - pergeometric series," Quart. J. Math. Oxford 1 9 (1968), 433-447. 6. G . E . Andrews, The Theory of Partitions, Addison-Wesley, Reading , 1976 . 7. G . E. Andrews, "An introduction to Ramanujan's 'lost' notebook," Amer. Math. Monthly 8 6 (1979), 89-108. 8. G . E. Andrews, q-series: Their Development and Application in Analysis, Num- ber Theory, Combinatorics, Physics, and Computer Algebra, CBM S Regiona l Conf . Ser. in Math. No. 66, Amer. Math. Soc, Providence, 1986 . 9. G . E . Andrews , "O n th e proof s o f th e Rogers-Ramanuja n identities, " i n q-Series and Partitions (D. Stanton, ed.), Springer-Verlag, New York, 1989 , pp. 1-14 . 10. G . E . Andrews , "Moc k thet a functions, " i n Theta Functions Bowdoin, Part 2, Proceedings of Symposia in Pure Mathematics, vol . 49 , America n Mathematica l Society, Providence, Rhode Island, 1989 , pp. 283-298. 11. G . E . Andrews , R . A . Askey , B . C . Berndt , K . G . Ramanathan , an d R . A . Rankin, eds., Ramanujan Revisited, Academic Press, Boston, 1988 . 12. G . E . Andrews, B . C. Berndt , L . Jacobsen, an d R . L . Lamphere, "Variation s on the Rogers-Ramanujan continue d fraction i n Ramanujan's notebooks, " in Number Theory, Madras, 198 7 (K . Alladi , ed.) , Lecture Note s i n Math . No . 1395 , Springer- Verlag, Berlin, 1989 , pp. 73-83 . 13. G . E. Andrews, B. C. Berndt, L. Jacobsen, and R. L . Lamphere, The continued fractions found in the unorganized portions of Ramanujan's notebooks, Mem. Amer . Math. Soc. 99 (1992), no. 477. 14. R . A. Askey, "Beta integrals in Ramanujan's papers , his unpublished work and further examples, " i n Ramanujan Revisited (G . E . Andrew s e t al. , eds.) , Academi c Press, Boston, 1988 , pp. 561-590. 15. R . A. Askey, "Beta integral s and ^-extensions, " Proc. Ramanujan Centennial International Conf, 15-1 8 December 1987 , Annamalainagar, Th e Ramanujan Math - ematical Soc, Madras, 1988 , pp. 85-102 .

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252. I . Schur, Gesammelte Abhandlungen, Ban d 2, Springer-Verlag, Berlin, 1973 . 253. J.-A . Seguier, La Thiorie des Formes Quadratiques et de la Multiplication Com- plexes Gauthier-Villars, Paris, 1894 . 254. A . Selberg, "Uber die Mock-Thetafunktionen siebente r Ordnung," Arch. Math. og Naturvidenskab 41 (1938), 1-15 . 255. A . Selberg , Collected Papers, Vol. 1 , Springer-Verlag , Berlin , 1989 , pp. 695- 706. 256. A . Selber g and S . Chowla, "O n Epstein' s zeta-functio n (I), " Proc. Nat. Acad. Sci. USA 35 (1949), 367-370. 257. A . Selberg and S . Chowla, "On Epstein's zeta-function," J. Reine Angew. Math. 227(1967), 86-110. 258. S . P. Sen, Dictionary of National Biography (4 volumes), Institute of Historical Studies, Calcutta, 1972-1974 . 259. N. M. Shah and B. M. Wilson, "On an empirical formula connected with Gold- bach's theorem," Proc. Cambridge Philos. Soc. 19 (1919), 238-244. 260. E . Shils, "Reflections o n tradition, centre and periphery an d the universal va- lidity o f science : th e significanc e o f the lif e o f S . Ramanujan," Minerva 2 9 (1991) , 393-419. 261. R . Sitaramachandrarao , "Som e formula e o f S . Ramanuja n III, " J. Madras Univ., Ser. B 51 (1988), 67-70. 262. S . Skewes , "O n th e differenc e n(x) - l i x (I), " Proc. London Math. Soc. 8 (1933), 277-283. 263. P . K . Srinivasan , Ramanujan: An Inspiration, Vols . 1 , 2 , Muthialpe t Hig h School, Madras, 1968 . 264. G . Szego, "Uber einige von S. Ramanujan gestellte Aufgaben," /. London Math. Soc. 3 (1928), 225-232. 265. G . Szego , Collected Papers, Vol . 2, Birkhauser, Boston , 1982 . 266. J . Tanner y an d J . Molk , Elements de la Thiorie des Fonctions Elliptiques ( 4 volumes), Gauthier-Villars, Paris, 1893-1902 . 267. E . C. Titchmarsh, Theory of Fourier Integrals, second ed. , Oxfor d Universit y Press, Oxford, 1948 . 268. H . W . Turnbull , "Bertra m Marti n Wilson, " Proc. Roy. Soc. Edinburgh 5 5 (1936), 176-177 . 269. R . C . Vaughan , The Hardy-Littlewood Circle Method, Cambridge Universit y Press, Cambridge, 1981. 270. K . Venkatachaliengar, Development of Elliptic Functions According to Ramanu- jan, Tech . Report 2, Madurai Kamaraj University , Madurai, 1988 . 271. T . Vijayaraghava n an d G . N. Watson , "Solutio n t o Questio n 784, " J. Indian Math. Soc. 19(1931), 12-23 . 272. S . S . Wagstaff , "Ramanujan' s pape r o n Bernoull i numbers, " J. Indian Math. Soc. 45(1981), 49-65. 273. K . C. Wali, Chandra, The University o f Chicago Press, Chicago, 1991. 274. G . N. Watson, A Treatise on the Theory ofBesset Functions, Cambridge Uni- versity Press, Cambridge, 1922 . 275. G . N. Watson, "Theorems stated by Ramanujan (II) : Theorems on summatio n of series," J. London Math. Soc. 3 (1928), 216-225. 276. G . N. Watson , "Theorem s state d b y Ramanuja n (IV) : Theorems o n approxi - mate integration and summation o f series," /. London Math. Soc. 3 (1928), 282-289. 277. G . N . Watson , "Solutio n t o Questio n 722, " J. Indian Math. Soc. 1 8 (1929) , 113-117. 278. G . N . Watson , "Theorem s state d b y Ramanuja n (V) : Approximation s con - nected wit h e x , " Proc. London Math. Soc. (2) 2 9 (1929), 293-308 . 334 REFERENCES

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Abel, R, 1 7 Abel-Plana summatio n formula , 40-4 1 Aberdeen University, 28 8 Accountant General , Madras, 1 Acta Mathematica, 14 6 Ananda Rau, K., 106 , 110-112 , 118 , 259-261, 264 , 266, 289, 290 Anantharaman, M. , 128 , 13 6 Andhra University, 27 7 Andrews, G.E., 42, 43, 65, 66, 224, 267, 304, 305 Apollonius Circle, 11 3 Arber, E.A.N., 15 4 Askey, R.A., 37 , 65, 313 Association o f Mathematics Teachers o f India, 27 1 Atkin, A.O.L., 194,29 7 Ayer, A.J., 45 Ayyar, M.V., 28 3

Bahadur, 10 6 Bailey, W.N., 40, 44, 66, 12 9 Baker, H.F., 2, 15 5 Balasubramanian, R. , 42, 19 9 Bapatla, 125 , 12 6 Barclays Bank, 234, 235, 237, 241 Barnes, E.W., 50 , 77, 82, 105 , 129 , 132 , 13 3 Barrow, I., 7 Bauer, G., 38 , 43 Berndt, B.C., 255, 285, 290, 304, 306, 307, passim Bernoulli numbers, 34 , 38-39, 49, 54, 63, 206, 313 Berry, A., 49, 77, 82 Berwick, W.E.H., 138-140 , 286 Bessel functions, 28 3 Bharathi, R. , 15 4 Bhargava, S. , 67

335 336 INDEX Bhujanga Rao , Dr., 21 4 Bialek, P., 19 1 Binet, A., 3 8 Birch, B.J., 19 4 Birmingham University, 282, 284, 288, 289, 299 Bollobas, B., 17 4 Bolton, Dr., 233, 234 Bombay, 12 6 Borwein, J.M. and P.B., 67, 19 9 Bose, J.C., 16 1 Bourne, A., 13 , 157, 207 Boyd, J.W.H., 17 3 Bradley, F.H., 11 5 Bromwich, T.J.I'A., 16-19 , 46, 53, 155, 208 Brouncker, Lord, 43 Buell, D., 14 0 Burkill, J.C., 30, 51

Cambridge University Press, 263 Cardiff, Universit y Colleg e of, 28 6 Carlson-Wigert, 282 , 284, 286, 287 Carr, G.S., 1 , 5 Cartwright, M.L., 5 1 Cathedral Road, Madras, 214 Cauchy, A.L., 38, 39, 10 6 Cauchy-Poisson formula , 279-28 1 Cauchy's theory o f residues, 46 Cauvery River, 215, 229 Cayley, A., 289, 290 Cazalet, V.A., 237, 238 Chan, H.H., 64 , 66, 14 1 Chandrakantha, Smt, 11 3 Chandrasekhar, P.S. , 306, 307, 310, 311 Chandrasekhar, S. , 193 , 255, 261, 290-294, 298, 306, 307, 309-31 1 Chandrasekharan, K. , 26 1 Chasles, ML , 230 Chebychev, P.L., 3 4 Chetpet, 219 , 224, 231 Chettiar, T.A. , 25 2 Chetty, P. Adinarayana, 227-23 0 Choultry, 24 5 Chowla, S. , 140 , 297 Chowry-Muthu, Dr. , 173 , 233, 234 circle method, 4 2 INDEX 337

Clark, E.M., 234, 235, 237, 241, 242 Cohen, H., 22 4 Coimbatore, 215, 216 Colinette Road, Putney, 174 , 201, 234 Cooum River, 5 2 Cottrell, C.B., 98, 10 0 Cours d'Analyse, 3 1 Course of Pure Mathematics, 3 2 Cranleigh, 3 0 Cromwell Road, London, 109 , 110 , 114 , 162 , 164 , 17 3 Cuddalore, 16 4 Cumbum, 7 3 Cursetjee, A. , 16 1

Darling, H.B.C., 29 8 Davenport, H. , 19 8 Davies, A., 11 , 70, 76, 86, 87, 99, 204, 209-211 Dedekind eta-function, 42 , 19 4 definite integrals , 24-26, 28, 47, 57-59, 116 , 12 0 de la Vallee-Poussin, C.J. , 8 7 Deshmukh, CD., 23 0 Deshouillers, J.-M., 19 9 Dewan, 10 6 Dewsbury, R, 51 , 53, 70, 76 , 78 , 80 , 94 , 96 , 100 , 129 , 132-136 , 142-144 , 156, 159 , 160 , 193 , 199 , 201, 202, 218, 224, 242, 263-266, 274 diets, 224 Dirac, P.A.M., 30 8 Dirichlet L-function , 40 , 281 Dirichlet, P.G.L., 19 9 Dirichlet series , 194 , 260, 261, 299 Dougall, J., 3 8 Dragonette, L. , 224 Dress, F., 19 9 Dundee, 26 8 Durairajan, Mr. , 12 4 Dutka, J., 4 3

Eddington, A.S., 15 4 Edinburgh University, 31 5 Educational Times, 7 5 Eisenstein series , 176-186 , 188-191 , 194 , 195 , 299, 300 Ekanatha Rao, S. , 11 3 elliptic functions, 19 , 96, 116 , 182 , 206, 289, 290, 314 elliptic integrals, 140 , 14 1 338 INDEX

Erode, 1,215,216 , 313-31 5 Euler-Maclaurin summatio n formula , 33 , 46 Euler products, 194 , 300, 30 1 Everest, Mount, 3 3

Fawcett, P., 8 Field, R.E., 171 , 17 3 First Arts Examination, 1 , 76, 144 , 206, 31 4 Fitzroy House, 3, 174 , 192-19 9 Forsyth, A.R., 15 5 Fort St George, 15 , 78, 261 Fourier integral, 28 6 Fourier transform, 28 0 Fowler, R.H., 26 2 Francis, W, 102 , 10 3 Frater, A., 5 3 Freeman-Thomas, Freeman , 238, 241

Ganapathy Iyer , C.N., 120-12 3 Ganapathy, M.M. , 128 , 13 6 Garvan, F.G., 6 7 Gauss, C.F., 123 , 13 9 Gibraltar, 23 8 Glaisher, J.W.L., 38 Glasgow University, 302 , 30 5 Godavery, 12 0 Goldberg, L.A. , 4 2 Gometra, 23 1 Gottingen, 24 3 Government Colleg e of Kumbakonam , 1 , 5, 93, 98, 158 , 247, 269, 289, 290, 313 Govindarajan, Mr. , 12 4 Grace, J.H., 15 5 Graham, J.F., 13 , 14 , 207,213 Gram, J.P., 34 , 63 Greenhill, A.G. , 138 , 140 , 289, 290 Griffin, N. , 4 5 Griffith, C.L.T. , 2 , 12-17 , 19 , 207 Guindy, 214, 215,24 1 Gupta, H. , 29 7

Hadamard, J., 8 7 Hafner, J.L. , 3 6 Halcrow, A., 30 4 INDEX 339

Hall, S. , 30 , 14 2 Hanumantha Rao , B., 70, 73, 76, 209 Hardy, G.E., 14 2 Hardy, G.H., biography, 30-33, passi m Hardy, I., 3 0 Harley Street, London, 21 5 Harry Ransom Humanitie s Center, 4 5 Hausman, M. , 3 5 Hecke, E., 194 , 301 Hecke zeta-function, 187 , 18 8 Heffer, W. , 229, 230 Heilbronn, H„ 243 , 299-301 Hensman, J J., 231 , 244 hepatic amoebiasis, 3 , 308, 309 Hermite, C, 138-14 0 Hickerson, D., 22 4 highly composite numbers, 138 , 286, 287 Hill Grove, 173 , 174,23 4 Hill, M.J.M., 2 , 13 , 15-19, 49, 208, 209 Hindu Hig h School , Kumbakonam, 73 , 292 Hirschhorn, M. , 6 5 Hobson, E.W. , 2 , 32, 15 5 Holland, T., 15 7 Home, W.O., 10 3 Huband, G.D. , 23 8 Hutton's Mathematical Tables , 232, 233 Huxley, M.N., 3 6 hypergeometric functions , 40 , 59 , 66, 67

Indian Mathematical Club , 74, 75, 10 6 Indian Mathematica l Society , 74 , 15 8 Ingham, A.E., 3 2 Ivic, A., 3 6 Iwaniec, H., 3 6

Jacobi, C.G.J., 41, 69, 10 6 Jacobsen, L. , 43, 65, 66 Janaki, S . (Mrs. Ramanujan) , 1 , 10 , 238-241, 248 , 252, 255-257, 261, 292, 293,306, 31 0 annuity to, 249, 252, 253, 255-257 Jeans, J.H., 31 , 153, 154 , 262 Journal of the Indian Mathematical Society, 1 , 17 , 34 , 39-41 , 70 , 75 , 116 , 117, 128 , 253, 265, 270, 272, 288, 289 Journal of the London Mathematical Society, 288 , 289 340 INDEX

Kanigel, R., 30 , 93, 193 , 216, 230 Kaplan, P., 14 1 K.C.I.E., 13 5 KesavaPai, Dr. , 214 Kiepert, L., 138 , 14 0 Kilpauk, 309 , 310 Kincaid, R, 153 , 168 , 169 , 173 , 233, 234 Klein, R, 138 , 13 9 Knopp, M.I., 193 , 194 , 297 Knorr, W.R., 23 0 Kodaikanal, 21 8 Kodumudi, 214, 215, 229, 230 Komalatammal, 1 , 10 , 118-119, 157 , 235, 236, 244-247, 249-251, 273-274, 310,315 Kratzel, E., 36 Kreczmer, W., 298 Krishnamachari, C.? , 127 , 12 8 Krishna Rao, R., 1 , 106-107 , 109-112 , 260 Krishnaswamy Iyer, A., 11 3 Kronecker, L., 13 8 Kumbakonam, 1 , 106, 126, 158 , 205, 217, 219, 229, 232, 245, 247, 250, 272, 291,313,315 Kumbakonam High School, 1 , 4, 97 , 102 , 113 , 246, 24 8

Lagrange inversion formula, 3 7 Lahore, 291 Lakshmi Narasimhan, S. , 231-233, 244, 246, 247, 273, 274, 314, 315 Lambert series, 290 Lamphere, R., 43, 65, 66 Landau, E., 32, 36, 42, 176 , 182 , 199 , 243, 244, 300 Laplace, P.S., 41 Larmor, J., 15 5 Laurence, R.V., 14 3 Lehigh University, 29 5 Lehmer, D.H., 194 , 196 , 295-29 7 Lehner, J., 299 Lerch, M., 3 8 Littlehailes, R. , 13 , 14 , 87 , 96 , 98 , 99 , 101 , 102 , 122 , 204 , 209 , 210 , 247 , 248, 274 Littlewood, J.E., 3, 32, 34, 35, 42, 44, 47, 48, 50 , 51, 68, 70, 76, 77, 81, 82, 89, 91 , 105 , 112 , 121 , 123 , 152 , 154 , 155 , 192 , 198 , 211, 242, 259, 260, 282, 290, 291 Livens, G.H., 285 , 286 Liverpool University, 268 , 289 INDEX 341

Lloyd George, David, 10 0 London Mathematica l Society , 18 , 70 , 112 , 113 , 122 , 131 , 140 , 151 , 224, 260, 287, 299, 302, 305 Loney, S.L. , 206 , 313 Host notebook', 182 , 191, 223, 224, 261, 267, 283, 293, 305 Love, A.E.H., 3 1

MacDonald, Ramsay , 12 9 MacMahon, P.A. , 141 , 142, 154 , 155 , 192 , 297 Macphail, E.M., 264, 265 Madgshon, J.B., 17 3 Madhava Rao, T., 106 , 11 3 Madras Central Station , 22 9 Madras Mail, 105 , 106 , 156 , 203, 227-230, 25 2 Madras Port Trust, 2 , 7-12, 21, 51, 53, 71, 73, 74, 78-81, 90 , 97, 122 , 128, 129, 143 , 204, 207, 209, 213, 217, 219, 255, 314 Madras Times, 203-213, 253, 254, 313 Madras University, 2 , 10 , 80, 95, 97-99, 102 , 105 , 113 , 128 , 131 , 133-135, 156, 183 , 201, 203-205, 210, 213, 232, 234, 238, 239, 252, 253, 257, 259, 263-265, 271, 273, 285, 286, 313, 315 Madurai, 7 3 MahaRaja Rajashry , 11 , 86 Mallet, C.E. , 90 , 103-105 , 209 Malurkar, S.L. , 39 , 283 Mambalam, 238 , 241, 245, 261 Mathematical Association, 9 6 mathematical tripos, 7, 8 Mathematician's Apology, 30, 32 Mathews, G.B., 138 , 13 9 Matlock House , 3 , 153 , 154 , 161 , 162 , 166 , 172-175 , 185 , 188 , 189 , 191 , 193, 233 McLean, W., 5 3 McVittie, G.C., 307 , 308 Medical Register, 219 Mellin's integral, 278, 280-282, 284, 286 Mexborough, 168 , 17 4 Middlemast, E.W. , 7 , 11 , 13, 15, 122, 206-208, 31 4 Minakshisundaram, S. , 261 Minkowski, H., 24 3 Mittag-Leffler, G. , 146 , 269 Mobius function, 6 3 mock theta functions , 220-224 , 26 0 modular equation, 60-62 , 67, 68, 116 , 138 , 139 , 290 Molk, J., 7 0 342 INDEX

Montagu, Mr. , 14 7 Mordell, L.J., 49 , 151 , 152, 194 , 287, 298, 299, 301 Morrell, Lady Ottoline, 44, 45 Mozzochi, C.J., 3 6 Mussolini, Benito, 3 3 Mutlin, Dr. , 23 3 Mysore, 13 2

Nan-Yue, Z., 14 1 Narasimha Iyengar , K. , 31 5 Narayana Aiyar , S. , 2 , 11 , 34, 49, 73-75, 79, 80, 86, 87, 93, 128 , 130 , 131, 133-135, 138 , 143 , 144 , 156 , 157 , 207-209, 211-213, 217, 219, 253-255 Narayanan, S. , 96 Narayanan, W. , 25 7 Narayaniengar, M.T. , 9 6 National Archives, Delhi, 130 , 131 , 133, 313 Nature, 153,274,27 5 Nehru, Jawaharlal, 25 5 Nellore, 1 , 106 , 131 , 206,209 nested radicals , 275-27 8 Neville, E.H., 2 , 89-91, 94-99 , 101 , 103-105, 109-111 , 114 , 121 , 204, 210, 211,233,274,275, 301 , 315 Neville, R., 11 0 New India, 213 , 214 Newton, I. , 7,44, 103 , 213 Nicholson, J.W., 15 5 Nicolas, J.-L., 28 7 Nilgris, 23 3 Nungambakkam, 52 , 96 Nursing Hostel, Thompson's Lane , Cambridge, 145 , 14 7

Offord, A.C. , 5 1 Old College, Madras, 1 4 Olver, F.W.J., 41 Ootacamund, 131-132 , 214, 216, 233, 247 Orders of Infinity, 21 , 12 3 Ostrowski, A.M., 3 5

Pachaiyappa's College , 1 , 6, 14 , 93, 113 , 114 , 118 , 314 Paley, R.E.A.C., 5 1 Palghat, 15 8 panchangam, 12 6 Parseval's theorem, 4 0 Partitio Numerorum, 259 , 260 INDEX 343 partition functio n p(n), asymptotic formula, 41 , 42, 141 , 142, 145 , 146 , 295, 296 congruences, 192-196 , 295-29 9 Patrachariar, K. , 122 , 12 3 Pentland, Lord, 100 , 211 Perron, O., 43, 66 Phillips, S.M. , 233 , 234 Pires, P.A., 21 6 Poisson summation formula , 40 , 283 Polya, G., 3 2 Preece, C.T., 37 , 38, 40, 43, 65-67, 288 Presidency College , Madras, 14 , 75, 96, 101 , 122 , 157 , 257, 259-261, 264 , 269,273,289,292, 307 , 314 Principia Mathematica, 4 5 Proceedings of the London Mathematical Society, 140 , 253, 255, 260 Purvis, J.E., 15 4 Putney, 3

Rademacher, H. , 42, 297, 305 Rajagopal, C.T. , 26 1 Ram, L., 162 , 166-168 , 171-174 , 233, 234 Ramachandra, K. , 4 2 Ramachandra Rao , R. , 1 , 2, 6 , 19 , 106 , 118 , 131 , 147 , 200, 206 , 208-209 , 214-215, 218, 232, 252-253, 255, 259, 260, 264, 265, 270, 291, 314 Ramalingam, A.S. , 161-174 , 23 4 Raman, C.V. , 122 , 161, 291, 292 Ramanathan, K.G. , 43, 64, 66, 68, 18 8 Ramaniah, R. , 15 8 Ramanujachari, C, 10 3 Ramanujaha, 10 1 Ramanujan, cause of death, 308 , 309 Collected Papers, 37, 43, 63, 67, 68, 80, 223, 242, 253, 263, 265, 272, 288, 291, 292, 298, 301 copyright o f papers, 249 degree from Cambridge , 131 , 136-137, 13 8 diets during his last months, 224-22 5 Fellow of the Royal Society, 152-157 , 159 , 161 , 192 Fellow of Trinity College , 3, 192 , 193 , 201, 234, 242, 243 medical accounts, 233-23 4 medical reports, 214-21 6 notebooks, 259, 264-266, 269, 270, 272, 285, 288, 310, 31 1 passport photo, 292-294, 29 8 postage stamp, 31 4 344 INDEX

quarterly reports, 80, 265 scholarship to Cambridge, 96, 98, 100-103 , 128 , 131-136 , 142 , 143 , 159, 160,202 scholarship to University of Madras, 78-81, 8 6 tau-function, 195 , 283 theory o f prim e numbers , 22 , 23 , 33 , 34 , 48 , 49 , 54-56 , 63 , 64 , 68 , 69 , 76, 77 , 81-89, 91-9 3 Ramanujan Institute, Madras , 277 Ramanujan Museum, Royapuram , 12 2 Ramaswami Aiyar , V., 75, 96, 206, 31 4 Ramaswami Iyer , C.P., 11 3 Ramunni Menon , K. , 249 , 256, 257 Ranganathan, A., 306 , 307 , 309-31 1 Ranganathan, S.R. , 201 , 230, 311, 315 Rangaswami, T.V. , 203 , 224 Rankin, R.A., 194 , 267, 287, 294, 302-30 5 Rao, M.B., 283 Reading, 96, 27 4 Riemann, B. , 34 , 12 3 Riemann hypothesis, 5 0 Riemann zeta-function, 17 , 44, 63, 68, 77, 116 , 188 , 278-281 Riesz, M. , 32 , 260 Roberts, S.C., 29 4 Robin, G., 28 7 Rogers, L.J., 43, 64, 220, 298 Rogers-Ramanujan continue d fraction , 29 , 43, 57 , 64, 66, 77, 87 , 90 Rogers-Ramanujan identities , 43, 65, 24 4 Rogosinski, W.W. , 3 2 Ross, E.B. , 206, 314 , 31 5 Royal Society, 3 , 32, 51 Royapuram, 24 5 Rushforth, J.M. , 19 6 Russell, B. , 44, 45, 291 Russell, Dr. , 218, 219 Russell, R. , 13 8

Sadagopan, Ms., 315 Saidapet, 238, 24 1 St Andrews University, 269 , 28 6 SankaraRao, Mr. , 11 8 Sarangapani Temple, 217, 21 8 Sarker, R.P., 5 3 Sastri, P.P.S., 23 0 Satagopan, T.A., 80 , 265, 30 2 INDEX 345

Satyamangalam, 7 5 Scholfield, A.F., 43, 244, 301, 302 Schroter, H., 138 , 14 0 Schur, I., 243, 244 Schwarz, W., 243 Seguier, J.-A., 13 9 Selberg, A., 42, 140 , 224, 297 Senate House, Cambridge , 13 7 Seshu Aiyar , P.V. , 106 , 112 , 118 , 157-159 , 235 , 236 , 244 , 245 , 253 , 255 , 259, 264-272, 29 1 Shah, N.M., 271,27 2 Shapiro, H.N., 3 5 Sharp, H., 257, 258 Shaw, H. Batty, 152 , 154 , 308, 30 9 Shils, E., 51 Sikorski, General, 23 8 Simla, 99 Sinclair, J., 100,21 1 Singaravelu Mudaliyar, S.P. , 117 , 11 8 singular moduli, 60 , 62, 67, 68, 288, 28 9 singular series, 147-15 2 Sitapati Ayyar, Mr., 214, 215 Sitaramachandrarao, R. , 3 9 Sivasailam, A., 205 Skewes, S., 12 3 Snow, C.P., 3 0 Sohncke, L.A. , 138 , 14 0 Soldner, J., 63 Spring, SirF., 2,12, 13 , 15 , 19 , 75, 79, 80, 87 , 98-100, 130 , 133-136 , 156 , 157, 162 , 207-209, 211-213, 253-25 5 Srinivasa Aiyangar, K. , 1 , 119 , 120 , 143 , 144 , 205, 247, 255, 31 4 Srinivasa Iyangar, R. , 245 , 252, 261 Srinivasa Rao, K. , 113 , 203, 224, 255 Srinivasan, P.K. , 52 , 78, 90, 106 , 119 , 122 , 230, 271 Srinivasan, P.S., 27 1 Stanley, A.E., 15 4 Stanley, G.K. , 286-28 8 Subbanarayanan, N., 7 5 Subramanian, L.V. , 15 8 Subramanian, S.M. , 116 , 123-128 , 136 , 146-14 7 Surya Narayana, T., 116-11 8 Suryanarayanan, S.S. , 113-115 , 117 , 14 7 Swaminathan, C.G. , 12 2 Szego, G., 4 1 346 INDEX

Tamil Nadu Archives, 70, 76, 100 , 215, 249, 250, 308 Tanjore, 11 3 Tannery, P., 70 Thacker, M.S., 255 Thomson, J.J., 152-154 , 242 Tirunarayanan, S. , 119 , 233, 246, 248, 273, 315 Tiruva Aeswaranpet, 7 3 Tiruvanakoil, 29 1 Titchmarsh, E.C., 30, 32 Trinity College, Cambridge , 2, 3, 30, 31, 45, 46 Trinity College, Dublin, 1 3 Triplicane, 73, 157 , 235, 244, 273

Vacation Lectures Fund, 101 , 103 , 132 , 202, 256 Vaidyanathaswamy, R. , 284-28 6 Vaughan, R.C., 42, 19 9 Venkatachaliengar, K., 6 7 Venkata Vilas, 214 Vijayaraghavan, T., 275-278, 288, 289 VinayakaRow, E. , 113-117 , 14 7 Vinogradov, I.M. , 42 Vizagapatnam, 1 Von Neumann, J. , 297 Vythynathaswamy, R. , 26 0

Wagstaff, S.S. , 1 7 Wali, K.C., 292, 310 Walker, Si r G., 2 , 51-53, 75, 76, 81, 86, 157 , 204, 209 Walker's quasi-biennial oscillation , 5 2 Waring's problem, 152 , 196-199 , 25 9 Watson, E.G., 267, 302, 304, 305 Watson, G.N., 36 , 38-41, 43, 44, 49, 64, 66-68, 80 , 194 , 196 , 224, 263, 265, 267, 269, 278, 282-289, 295 , 297-305 Watson, G . T., 286 , 30 5 Webb, R.R., 3 1 Weber, H., 67 , 289 Wells, 3 Westfield College , London, 287 Whipple, F.J.W., 66 Whitehead, A.N., 45, 15 5 Whittaker, E.T., 155 , 283, 305, 307 Whittaker, J.M., 304 , 30 5 Williams, K.S., 14 1 Wilson, B.M., 267-272, 274, 275, 284, 285, 288, 289, 302-30 4