Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence

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.

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 . - 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/ . - 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 -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. -