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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.