This authoritative volume covers aspects of the life and enduring mathematical research of Srinivasa Ramanujan. Born in the late 19th century, Ramanujan had almost no formal training in pure mathematics. This iconic figure made extraordinary contributions to many aspects of mathematical analysis and number theory. During his short life, Ramanujan published 37 papers and curated in several 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 pure mathematics and theoretical physics. The eight editors of this Handbook have assembled details on all 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.
Preface.- Part. I. Life and History - Life and Family.- Srinivasa
Ramanujan.- The name, Srinivasa Ramanujan.- S. Janaki Ammal (Mrs.
Ramanujan).- The Ramanujan family record.- Ramanujans father, Kuppuswamy
Srinivasa Aiyangar.- Ramanujans brothers, Lakshmi Narasimhan and
Tirunarayanan.- Ramanujans 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, Ramanujans
hometown.- Ramanujans 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.- Pachaiyappas College.- The Sarangapani Temple.- The Ramanujan Math
Park.- Part III Life and History - Ramanujans Books .- Carrs
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.- Ramanujans quarterly reports.- Ramanujans last letter to
Hardy.- Ramanujans 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.- Ramanujans 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.- Ramanujans
Notebooks.- Ramanujans Notebooks Part I.- Ramanujans NotebooksPart
II.- Ramanujans Notebooks Part III.- Ramanujans Notebooks Part
IV.- Ramanujans Notebooks Part V.- The discovery of Ramanujans Lost
Notebook.- Ramanujans Lost Notebook Part I.- Ramanujans Lost Notebook Part
II.- Ramanujans Lost Notebook Part III.- Ramanujans Lost Notebook Part
IV.- Ramanujans Lost Notebook Part V.- Your hit parade: The top ten most
fascinating formulas in Ramaujans Lost Notebook.- Part VIII. Legacy Books
and Journal Articles.- A Mathematicians 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.- Ramanujans Place in the World of
Mathematics: Essays providing a comparitive study.- Resonance of Ramanujans
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 (childrens
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.- Indias 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 .- 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.- Ramanujans continued fraction involving gamma
functions.- Ramanujans paper on Riemanns functions (s) and (t) and a
transformation from the Lost Notebook.- Ramanujans series for 1/
.- Ramanujans taxi cab number.- Ramanujans work on Euler
products.- Ramanujans 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.- Baileys 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.- Nahms Conjecture.- Ramanujan and Computer
Algebra.- Ramanujan and Heines method.- Ramanujan and hypergeometric
series.- The Rogers-Ramanujan continued fraction.- Evaluations of the
Rogers-Ramanujan continued fraction.- Ramanujans continued fractions
involving hypergeometric functions The Rogers-Ramanujan continued fraction.-
Ramanujans general q-continued fractions.- The 1 1 summation.- Watsons
q-Whipple transformation formula.- Part XIII Mathematics -
Integrals.- Beautiful integrals.- Elliptic and related integrals.- Mordell
integrals.- q-Beta integrals.- Ramanujans 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 AppellLerch
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 Ramanujans 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.- Dysons rank
and crank conjectures and partition congruences.- Identities for the
Rogers-Ramanujan functions.- Ramanujan and the Nekrasov-Okounkov
formula.- Ramanujans modular equations with applications to
partitions.- Ramanujans partition congruences.- Ramanujans 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 Ramanujans 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.- Ramanujans alternative theories of theta
functions.- Rankin-Cohen brackets.- Singular moduli.- Combinatorial formulas
for Ramanujans tau function.- The Ramanujan -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.- Ramanujans
master theorem.- Part XVIII Mathematics - Squares and Cubes.- Gausss circle
problem and Dirichlets 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.-
KRISHNASWAMI ALLADI is professor of mathematics at the University of Florida where he was department chairman during 19982008. He received his PhD from the University of California, Los Angeles, in 1978. His research covers many areas of number theory such as analytic number theory, diophantine approximations, sieve methods, probabilistic number theory, and the theory of partitions and q-hypergeometric series. He is the Founder and Editor-in-Chief of The Ramanujan Journal (Springer), Founder and Editor of the book series Developments in Mathematics (Springer) and Chair of the SASTRA Ramanujan Prize Committee. He was an Associate Editor of the Notices of the American Mathematical Society. In 2012 he was inducted as an Inaugural Fellow of the American Mathematical Society for his distinguished contributions. In recognition of his research accomplishments and service to the profession, he was awarded an Honorary Doctorate (Honoris Causa) by SASTRA University in September 2022. GEORGE ANDREWS received his PhD from the University of Pennsylvania in 1964. He was Hans Rademachers 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 20092011. In 1976, while visiting the Wren Library of Trinity College, Cambridge, he had the good fortune to find the manuscript now called Ramanujans 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 Ramanujans notebooks and his lost notebook. He published five books on Ramanujans 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 Ramanujans 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 19981999. 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 19831986 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, Ramanujans 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 Directors 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 (198486, 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-1990s to the mid-2010s. 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 20212022. 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.
