Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence
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This authoritative volume covers aspects of the life and enduring mathematical research of Srinivasa Ramanujan. Born in the late 19th century, Ramanujan had little formal training in pure mathematics. This iconic figure made extraordinary contributions to many facets of mathematical analysis and number theory. During his short life, Ramanujan published 37 papers and curated in notebooks more than 3900 identities which he recorded without proof. Nearly all of his claims that were new have now been proven correct. He stated numerous results that were both original and highly unconventional. Many of these identities have led to major achievements in a wide range of areas of mathematics and theoretical physics. The eight editors of this Handbook have assembled articles on many aspects of Ramanujan's life and mathematical legacy with a focus on the evolution of his discoveries into many important sub-disciplines of current mathematical research. Included are 234 articles supplied by 88 authors. The book will be of interest to students, teachers, researchers and anyone who is intrigued by the legacy of one of the most striking figures in the history of mathematics.
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GEORGE ANDREWS received his PhD from the University of Pennsylvania in 1964. He was Hans Rademacher's last student. He is the Atherton Professor of Mathematics at the Pennsylvania State University. He is a member of the National Academy of Sciences (USA), a fellow of the American Academy of Arts and Sciences, a fellow of the American Mathematical Society and a fellow of the Society of Industrial and Applied Mathematics. He holds honorary doctorates from the University of Parma (Italy), the University of Florida, the University of Waterloo (Canada), SASTRA University (India), and the University of Illinois. He was awarded an honorary professorship at Nankai University (China). He was a Guggenheim Fellow in 1983. In 2022, the Institute of Combinatorics and its Applications presented him with the Euler Medal. He was President of the American Mathematical Society from 2009-2011. In 1976, while visiting the Wren Library of Trinity College, Cambridge, he had the good fortune to find the manuscript now called Ramanujan's Lost Notebook. He and Bruce Berndt collaborated on a five volume explication of the unproven identities in this masterpiece.
BRUCE BERNDT received an A.B. degree in 1961 from Albion College, and a PhD in 1966 at the University of Wisconsin, where his primary mentors were Rod Smart (thesis advisor), Marvin Knopp, and Richard Askey. He held a one-year postdoctoral position at the University of Glasgow, where his mentor was Robert Rankin. In 1967, Berndt joined the faculty at the University of Illinois, where he taught for 52 years before retiring in 2019. While at Illinois, he served as the thesis advisor for 37 students; he also mentored about ten postdocs. Since 1974, Berndt has devoted the bulk of his research to explicating and proving claims from Ramanujan's notebooks and his lost notebook. He published five books on Ramanujan's Notebooks (Springer), for which he received the Steele Prize for Mathematical Exposition from the American Mathematical Society. Furthermore, he and George Andrews published five books on Ramanujan's Lost Notebook (Springer). Altogether, Berndt has authored, co-authored, and co-edited 30 books. He received the Ford Award (twice) and the Allendorfer Award for papers published with the Mathematical Association of America. Berndt is the founding editor of International Journal of Number Theory . He was a Guggenheim Fellow in 1998-1999.
FRANK GARVAN received his B.Sc.(Hons) and DipEd. from the University of New South Wales (UNSW), Australia. After he taught high school in inner city Sydney and Outback NSW he completed a M.Sc.(research) degree at UNSW under the supervision of Mike Hirschhorn. From 1983-1986 he did a PhD under the supervision of George Andrews at the Pennsylvania State University. He then was a post-doc at the University of Wisconsin (mentor Richard Askey), the Institute for Mathematics and Its Applications (mentor Dennis Stanton), Macquarie University (mentor John Loxton), and Dalhousie University (mentors Jon and Peter Borwein). Since 2000 he has been a full professor at the University of Florida. His interests include q-series, partitions, mock theta functions, and symbolic computation. He has written books on the symbolic software MAPLE and written numerous public packages to aid q-series research. He is Managing Editor of The Ramanujan Journal . He has been privileged to organize several number theory conferences in Florida together with their conference proceedings. In 2022, he was honored to be the principal lecturer for the CBMS Conference, Ramanujan's Partition Congruences, Mock Theta Functions, and Beyond , at the University of Texas Rio Grande Valley.
KEN ONO is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia. He is a Fellow of the Shannon Center for Advanced Studies. He earned his PhD in mathematics from UCLA in 1993, and he earned a BA in mathematics from the University of Chicago in 1989. He is a recipient of a Guggenheim Fellowship, David and Lucile Packard Fellowship, Alfred P. Sloan Foundation Fellowship, and an NSF CAREER Award. In 2000 he received a Presidential Early Career Award from Bill Clinton at a ceremony at the White House. In 2005 he was named the National Science Foundation Director's Distinguished Teaching Scholar, the highest honor awarded by NSF for excellence in research and teaching. He is a Fellow of the American Mathematical Society, and he is the recipient of the 2023 University of Chicago Alumni Medal for Professional Achievement. Heis the Founding Director of the Spirit of Ramanujan Global STEM Talent Search, which supports emerging engineers, mathematicians, and scientists who lack traditional institutional support through financial grants and mentorship opportunities. He was an Associate Producer of the film The Man Who Knew Infinity , which starred Dev Patel and Jeremy Irons.
PETER PAULE earned his doctorate from the University of Vienna in 1983 (advisor: Johann Cigler). He was awarded a Humboldt Research Fellowship (1984-86, University of Bayreuth, host: Adalbert Kerber). In 2005 he was appointed full professor as successor to Bruno Buchberger; from 2009 to 2023 he was director of the Research Institute for Symbolic Computation (RISC) of the Johannes Kepler University Linz (JKU). From 2003 to 2008 he was Speaker of the Special Research Program Numerical and Symbolic Scientific Computing, from 2008 to 2017 director of the Doctoral Program Computational Mathematics, both excellence programs at JKU sponsored by the Austrian Science Fund (FWF). Paule was a member of the original editorial committee for the Digital Library of Mathematical Functions (DLMF) project, in existence from the mid-1990's to the mid-2010's. He is adjunct professor at the Center for Applied Mathematics, Tianjin University. He is a member of the Academia Europaea, and a Fellow of the American Mathematical Society.
OLE WARNAAR completed his PhD at the University of Amsterdam in 1993 under the guidance of Bernard Nienhuis, and currently holds the position of Chair and Professor of Pure Mathematics at The University of Queensland, Australia. His research interests include algebraic combinatorics, basic and elliptic hypergeometric series, q-series and partition theory, representation theory, and special functions. Warnaar has held a Research Fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW) from 1997 to 2000, was elected Fellow of the Australian Academy of Science in 2008, and was awarded the George Szekeres Medal of the Australian Mathematical Society (AustMS) in 2020. He served as President of AustMS in 2021-2022. Warnaar has a long-standing involvement with Australian as well as international mathematics competitions, and chaired the organizing committee of the Simon Marais Mathematics Competition (SMMC) from 2019 till 2022.
AE JA YEE is professor of mathematics at the Pennsylvania State University. She received her PhD from Korea Advanced Institute of Science and Technology in 2000. Her research covers areas of combinatorics and number theory, in particular the theory of partitions and q-hypergeometric series. She is a managing editor of International Journal of Number Theory and an associate editor of Discrete Mathematics . She also serveson the editorial boards of The Ramanujan Journal and Combinatory Theory .
Content
Preface.- Part. I. Life and History - Life and Family.- Srinivasa Ramanujan.- The name, Srinivasa Ramanujan.- S. Janaki Ammal (Mrs. Ramanujan).- The Ramanujan family record.- Ramanujan's father, Kuppuswamy Srinivasa Aiyangar.- Ramanujan's brothers, Lakshmi Narasimhan and Tirunarayanan.- Ramanujan's illness.- On the discovery of the photograph of S. Ramanujan, FRS.- Komalattammal.- Namagiri, The Goddess of Namakkal.- Ramanujan bust.- Paul T. Granlund.- Part II: Life and History - Places.- Erode, the birthplace of Ramanujan.- Kumbakonam, Ramanujan's hometown.- Ramanujan's home in Kumbakonam.- Government Arts College, Kumbakonam.- Kumbakonam higher secondary school.- Madras Port Trust.- The Madras Port Trust and Sir Francis Spring.- University of Madras.- Pachaiyappa's College.- The Sarangapani Temple.- The Ramanujan Math Park.- Part III Life and History - Ramanujan's Books .- Carr's Synopsis.- Matriculation exams taken by Ramanujan.- The books studied by Ramanujan in India.- Part IV Life and History - Correspondences.- Introduction to the Correspondence between Ramanujan and Hardy.- S. Ramanujan to G. H. Hardy 16 January 1913.- G. H. Hardy to S. Ramanujan 8 February 1913.- S. Ramanujan to G. H. Hardy 27 February 1913.- G. H. Hardy to S. Ramanujan 26 March 1913.- Letters to G. H. Hardy from nursing homes.- Ramanujan's quarterly reports.- Ramanujan's last letter to Hardy.- Ramanujan's slate.- Part V Life and History - Institutions.- The Ramanujan Mathematical Society.- The Royal Society and its Fellowship.- Trinity College and its Fellowship.- SASTRA Deemed University.- Part VI Biographies.- S. Narayana Aiyar.- Richard A. Askey.- A. O. L. Atkin.- W. N. Bailey.- H. F. Baker.- Arthur Berry.- SubrahmanyamChandrasekhar.- Ramanujan's Physician, P. S. Chandrasekhara Iyer.- Sir Francis Dewsbury.- Freeman Dyson.- Leonhard Euler.- A. G. Greenhill.- G. H. Hardy.- M. J. M. HILL.- E. W. Hobson.- V. Ramaswami Iyer.- P. V. Seshu Iyer.- C. G. J. Jacobi.- Richard Littlehailes.- J. E. Littlewood.- P. A. MacMahon.- P. C. Mahalanobis.- Louis J. Mordell.- E. H. Neville.- Hans Rademacher.- R. Ramachandra Rao.- A. S. Ramalingam.- S. R. Ranganathan.- Robert A. Rankin.- L. J. Rogers.- Issai Schur.- Lucy Joan Slater.- Sir Francis Spring.- P. K. Srinivasan.- Swinnerton-Dyer and the Ramanujan congruences for _ (n).- V. R. Thiruvenkatachar.- K. Venkatachaliengar.- Sir Gilbert Walker.- G. N. Watson.- B. M. Wilson.- Part VII. Legacy - Notebooks.- Ramanujan's Notebooks.- Ramanujan's Notebooks Part I.- Ramanujan's NotebooksPart II.- Ramanujan's Notebooks Part III.- Ramanujan's Notebooks Part IV.- Ramanujan's Notebooks Part V.- The discovery of Ramanujan's Lost Notebook.- Ramanujan's Lost Notebook Part I.- Ramanujan's Lost Notebook Part II.- Ramanujan's Lost Notebook Part III.- Ramanujan's Lost Notebook Part IV.- Ramanujan's Lost Notebook Part V.- Your hit parade: The top ten most fascinating formulas in Ramaujan's Lost Notebook.- Part VIII. Legacy - Books and Journal Articles.- A Mathematician's Apology.- Honoring a gift from Kumbakonam.- Mathematics Wizard Srinivasa Ramanujan.- Number Theory in the Spirit of Ramanujan.- Ramanujan Memorial Number Vol. 1, Letters and Reminiscences: Vol. 2, An Inspiration, P. K. Srinivasan, ed.- Ramanujan - The Man and the Mathematician.- Ramanujan: Man of Mathematics (comic book).- Ramanujan: Twelve Lectures on Subjects suggested by his Life and Work, by G.H. Hardy.- Collected Papers.- Ramanujan's Place in the World of Mathematics: Essays providing a comparitive study.- Resonance of Ramanujan's Mathematics (in three volumes) Books by R. P. Agarwal.- Srinivasa Ramanujan - Life and Work of a Natural Mathematical Genius, Swayambhu.- The Boy Who Dreamed of Infinity: A Tale of the Genius of Ramanujan (children's book).- The final problem: an account of the mock theta functions.- The Man Who Knew Infinity - a life of the genius Ramanujan.- The Mathematical Legacy of Srinivasa Ramanujan.- The Ramanujan Journal.- Part IX. Legacy - Plays Movies and TV.- A First Class Man.- A disappearing number.- Documentaries on Srinivasa Ramanujan by Mr. Nandan Kudhyadi.- On the man who loved numbers.- Partition - A Play on Ramanujan.- Ramanujan - The Opera.- The Man Who Knew Infinity (Movie).- The Ramanujan CD.- Part X. Legacy - Conferences Events and Prizes.- The ICTP Ramanujan Prize.- The SASTRA Ramanujan Prize.- Some Ramanujan Centenary Conferences.- Ramanujan at 125.- The Ramanujan Trust in Mauritius.- India's efforts to commemorate Ramanujan including national mathematics day.- Part XI Mathematics - Classical.- Stirling and Bell numbers and related topics.- Bernoulli numbers.- Busche-Ramanujan identities.- Circular summation.- Class invariants.- Diophantine approximation.- Elementary algebraic identities and polynomial equations.- Highly composite numbers and other champion numbers.- Partial fractions.- Problems solved by Ramanujan in the Journal of the Indian Mathematical Society.- The Problems submitted by Ramanujan to the Journal of the Indian Mathematical Society.- Radicals.- Ramanujan and modular forms.- Ramanujan and odd zeta values.- Ramanujan and p.- Ramanujan and Probabilistic Number Theory.- Ternary quadratic forms.- Ramanujan and the Riemann zeta-function.- Ramanujan complexes.- Ramanujan Graphs.- Ramanujan summation in quantum field theory.- Ramanujan sums.- Ramanujan sums and the general linear group.- Ramanujan's continued fraction involving gamma functions.- Ramanujan's paper on Riemann's functions ?(s) and ?(t) and a transformation from the Lost Notebook.- Ramanujan's series for 1/ p.- Ramanujan's taxi cab number.- Ramanujan's work on Euler products.- Ramanujan's Question.- Signal processing and Ramanujan sums.- The 'Constant' of a series.-The Hardy-Littlewood-Ramanujan Method.- The normal number of prime factors of a number n.- G. N. Watson - 31 papers inspired by Ramanujan.- Missing works of Ramanujan.- Part XII Mathematics - q-Series.- The Rogers-Ramanujan identities.- The Rogers-Ramanujan identities: Antecedents.- Bailey's tranform and lemma.- Rogers-Ramanujan identities in the hard hexagon model.- A touch of quantum field theory.- K3 Surfaces and sums of two cubes.- Knot theory and Rogers-Ramanujan type identities.- Lepowsky-Wilson Z-algebra and Rogers-Ramanujan-type identities: recent advances.- Nahm's Conjecture.- Ramanujan and Computer Algebra.- Ramanujan and Heine's method.- Ramanujan and hypergeometric series.- The Rogers-Ramanujan continued fraction.- Evaluations of the Rogers-Ramanujan continued fraction.- Ramanujan's continued fractions involving hypergeometric functions The Rogers-Ramanujan continued fraction.- Ramanujan's general q-continued fractions.- The 1 1 summation.- Watson's q-Whipple transformation formula.- Part XIII Mathematics - Integrals.- Beautiful integrals.- Elliptic and related integrals.- Mordell integrals.- q-Beta integrals.- Ramanujan's published papers on definite integrals.- Trigonometric Dirichlet series and Eichler integrals.- Part XIV Mathematics - Mock Theta Functions.- A century of answering the question: What is a mock theta function?.- A survey article on Appell-Lerch functions.- False theta functions classical theory.- False theta functions and modular forms.- Harmonic Maass forms and mock modular forms.- Mock theta functions.- Partial theta functions.- Combinatorics of Ramanujan's third order mock theta functions.- Polar Harmonic Maass forms.- Quantum modular forms.- Ramanujan and quantum black holes.- Part XV Mathematics - Partitions.- An algebraic formula for the partition function.- Dyson's rank and crank conjectures and partition congruences.- Identities for the Rogers-Ramanujan functions.- Ramanujan and the Nekrasov-Okounkov formula.- Ramanujan's modular equations with applications to partitions.- Ramanujan's partition congruences.- Ramanujan's Unpublished Manuscript on the Partition and Tau Functions.- The G¨ollnitz-Gordon identities.- Overpartitions.- Part XVI Mathematics - Theta Functions Elliptic Functions and Modular Forms.- Automatic discovery and proofs of formulas of Ramanujan-Kolberg type.- Divisibility of fourier coefficients of modular forms.- Eichler cohomology: A view of Ramanujan's mock modular forms.- Eisenstein series I.- Eisenstein series II.- Elliptic functions.- Explicit values of theta functions.- Indefinite theta series.- The Langlands Program.- Modular Equations.- The modular j-invariant.- Modular forms and the modular group.- Monster denominator formula.- Monstrous moonshine.- Pariah moonshine.- Umbral moonshine.- Ramanujan's alternative theories of theta functions.- Rankin-Cohen brackets.- Singular moduli.- Combinatorial formulas for Ramanujan's tau function.- The Ramanujan t -function.- Theta functions.- Part XVII Mathematics - Analysis.- Bessel functions and summation formulae.- Gaussian quadrature and orthogonal polynomia.- The Euler-Maclaurin and Abel-Plana Summation Formulas.- Asymptotic expansions.- Ramanujan's master theorem.- Part XVIII Mathematics - Squares and Cubes.- Gauss's circle problem and Dirichlet's divisor problem.- Magic squares.- Number of numbers representable as a square or sum of two squares.- Ramanujan and sums of two cubes.- The Ramanujan-Mordell Theorem on sums of an even number of squares.-