The Mathematical Legacy of Srinivasa Ramanujan

Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan's essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan's significant papers and show how these papers shaped the course of modern mathematics.

List of contents

Preface.- Chapter 1. The Legacy of Srinivasa Ramanujan.- Chapter 2. The Ramanujan tau function.- Chapter 3. Ramanujan's conjecture and l-adic representations.- Chapter 4. The Ramanujan conjecture from GL(2) to GL(n).- Chapter 5. The circle method.- Chapter 6. Ramanujan and transcendence.- Chapter 7. Arithmetic of the partition function.- Chapter 8. Some nonlinear identities for divisor functions.- Chapter 9. Mock theta functions and mock modular forms.- Chapter 10. Prime numbers and highly composite numbers.- Chapter 11. Probabilistic number theory.- Chapter 12. The Sato-Tate conjecture for the Ramanujan tau-function.- Bibliography.- Index.

About the author

M. Ram Murty is head of the Department of Mathematics and Statistics at Queen's University, and is a Queen's research chair in mathematics. He is also professor of philosophy at Queen's. He was elected a fellow of the Royal Society of Canada in 1990, the Indian National Science Academy in 2008, and won the Coxeter–James Prize, Jeffery–Williams Prize, E.W.R. Steacie Fellowship, and the Killam Fellowship. His research areas include number theory, modular forms, elliptic curves, and sieve theory. His book Non-vanishing of L-functions and Applications, coauthored by his brother V. Kumar Murty, won the 1996 Balaguer Prize and was published by Birkhauser. In addition, Ram is adjunct professor at McGill University; TIFR; IMSc; CMI; IIT Bombay; IISER, West Bengal; Vivekananda University; and Harish Chandra Research Institute, Uttar Pradesh.
V. Kumar Murty is head of the Department of Mathematics at the University of Toronto. His research areas include number theory, arithmetic geometry, and their applications. He was awarded the Coxeter-James Prize of the Canadian Mathematical Society in 1991, the E. W. R. Steacie Fellowship of NSERC in 1995, and the Inventor of the Year Award of the University of Toronto in 2011. He was elected a fellow of the Royal Society of Canada in 1995 and a fellow of the National Academy of Sciences in 2011. He served as editor-in-chief of the Journal of the Ramanujan Mathematical Society, and as editor of the Transactions of the American Mathematical Society. Moreover, he served on the board of directors of the Canadian Mathematical Society during 1995–1999 and as vice-president of the society during 2009–2011. Kumar Murty received his PhD from Harvard University. He is adjunct professor at TIFR, IMSc, CMI, the Harish-Chandra Research Institute, and Vivekananda University.

Summary

Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan’s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan’s significant papers and show how these papers shaped the course of modern mathematics.

Additional text

“This book is based on a number of talks the authors gave at various research institutes. Their aim is to survey some of Srinivasa Ramanujan’s most significant achievements and the developments they have led to over the last decades. … This volume is suitable as a first introduction to some of Ramanujan’s remarkable and deep ideas.” (C. Baxa, Monatshefte für Mathematik, Vo. 180, 2016)
“The authors introduce the reader to the topics through a historical account of the origin and later developments. … We warmly recommend this book for those who would like to have a glimpse on Ramanujan’s mathematics. Without being lost in the technicalities the reader will get a good look at the shape of many central questions.” (Péter Hajnal, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)
“The Murtys’ goal … is to present Ramanujan’s mathematical legacy to a broad audience, and the thrust of the book is a set of eleven chapters discussing exactly that. … the book will utterly charm you, given its accessibility, style, structure, and depth. It’s a great pleasure to read, and it’s fine scholarship.” (Michael Berg, MAA Reviews, December, 2013)
“This book is an outcome of lectures given by both authors in the last quarter century on various aspects of Ramanujan’s work at research institutions in India and Canada, to audiences ranging from graduate students to senior researchers. Thus the book is written in a style that will appeal to both research mathematicians and students desiring to enter the exciting world of Ramanujan’s mathematics. … the exposition is good and the book will be of interest to students and researchers … .” (Krishnaswami Alladi, Mathematical Reviews, September, 2013)

Report

"This book is based on a number of talks the authors gave at various research institutes. Their aim is to survey some of Srinivasa Ramanujan's most significant achievements and the developments they have led to over the last decades. ... This volume is suitable as a first introduction to some of Ramanujan's remarkable and deep ideas." (C. Baxa, Monatshefte für Mathematik, Vo. 180, 2016)
"The authors introduce the reader to the topics through a historical account of the origin and later developments. ... We warmly recommend this book for those who would like to have a glimpse on Ramanujan's mathematics. Without being lost in the technicalities the reader will get a good look at the shape of many central questions." (Péter Hajnal, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)
"The Murtys' goal ... is to present Ramanujan's mathematical legacy to a broad audience, and the thrust of the book is a set of eleven chapters discussing exactly that. ... the book will utterly charm you, given its accessibility, style, structure, and depth. It's a great pleasure to read, and it's fine scholarship." (Michael Berg, MAA Reviews, December, 2013)
"This book is an outcome of lectures given by both authors in the last quarter century on various aspects of Ramanujan's work at research institutions in India and Canada, to audiences ranging from graduate students to senior researchers. Thus the book is written in a style that will appeal to both research mathematicians and students desiring to enter the exciting world of Ramanujan's mathematics. ... the exposition is good and the book will be of interest to students and researchers ... ." (Krishnaswami Alladi, Mathematical Reviews, September, 2013)

Product details

• Authors: Kumar Murty, M Ra Murty, M Ram Murty, M. Ram Murty, Ram Murty, V Kumar Murty, V. Kumar Murty
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 30.11.2012
• EAN: 9788132207696
• ISBN: 978-81-322-0769-6
• No. of pages: 188
• Dimensions: 175 mm x 16 mm x 245 mm
• Weight: 436 g
• Illustrations: XII, 188 p.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Algebra, B, History, Mathematics, Combinatorics, Mathematics and Statistics, Discrete Mathematics, Number Theory, Combinatorics & graph theory, History of mathematics, History of Mathematical Sciences, Special Functions, Functional analysis & transforms, Fourier Analysis