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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson.  Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988.  The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society.

Functions, Theta. --- Mathematicians -- India -- Biography. --- Ramanujan Aiyangar, Srinivasa, 1887-1920. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Mathematics - General --- Mathematicians --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Mathematics. --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of

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This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

q-series. --- Mathematics. --- Ramanujan Aiyangar, Srinivasa, --- Math --- Science --- Series --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Geometry, algebraic. --- Sequences (Mathematics). --- Functions, special. --- Algebraic Geometry. --- Sequences, Series, Summability. --- Special Functions. --- Special functions --- Mathematical analysis --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Special functions.

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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This fifth and final installment of the authors’ examination of Ramanujan’s lost notebook focuses on the mock theta functions first introduced in Ramanujan’s famous Last Letter. This volume proves all of the assertions about mock theta functions in the lost notebook and in the Last Letter, particularly the celebrated mock theta conjectures. Other topics feature Ramanujan’s many elegant Euler products and the remaining entries on continued fractions not discussed in the preceding volumes. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society.

Mathematical analysis. --- Mathematicians --- 517.1 Mathematical analysis --- Mathematical analysis --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Functions, special. --- Functions of complex variables. --- Number theory. --- Special Functions. --- Functions of a Complex Variable. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Complex variables --- Elliptic functions --- Functions of real variables --- Special functions --- Special functions.

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This book is a collection of articles, all by the author, on the Indian mathematical genius Srinivasa Ramanujan as well as on some of the greatest mathematicians in history whose lives and works have things in common with Ramanujan. It presents a unique comparative study of Ramanujan’s spectacular discoveries and remarkable life with the monumental contributions of various mathematical luminaries, some of whom, like Ramanujan, overcame great difficulties in life. Also, among the articles are reviews of three important books on Ramanujan’s mathematics and life. In addition, some aspects of Ramanujan’s contributions, such as his remarkable formulae for the number π, his pathbreaking work in the theory of partitions, and his fundamental observations on quadratic forms, are discussed. Finally, the book describes various current efforts to ensure that the legacy of Ramanujan will be preserved and continue to thrive in the future. Thus the book is an enlightening study of Ramanujan as a mathematician and a human being. From the Foreword by George Andrews—one of the greatest experts on Ramanujan's work: “Alladi, who has worked in several areas of number theory and analysis, and who, as editor of the Ramanujan Journal, is uniquely qualified to write these historical sketches which provide an unusual and compelling view of Ramanujan.” ABOUT THE AUTHOR Krishnaswami Alladi is professor of mathematics at the University of Florida, where he was the department chairman during 1998–2008. He received his PhD from the University of California, Los Angeles, in 1978. His research area is number theory, where he has made notable contributions. In 1987, during the Ramanujan Centennial in India, he got the inspiration to launch The Ramanujan Journal (now published by Springer), devoted to all areas of mathematics influenced by Ramanujan. He annually writes articles about Ramanujan and his place in the world of mathematics, for journals and newspapers. He is presently editor-in-chief of The Ramanujan Journal, editor of the book series Developments in Mathematics (Springer), and associate editor of Notices of the American Mathematical Society.

Mathematics -- Popular works. --- Mathematics. --- Quadratic forms. --- Mathematicians --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Mathematics - General --- History --- History. --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Number theory. --- Mathematics, general. --- Number Theory. --- History of Mathematical Sciences. --- Scientists --- Number study --- Numbers, Theory of --- Algebra --- Math --- Science --- Annals --- Auxiliary sciences of history --- India. --- Bharat --- Bhārata --- Government of India --- Ḣindiston Respublikasi --- Inde --- Indi --- Indien --- Indii͡ --- Indland --- Indo --- Republic of India --- Sāthāranarat ʻIndīa --- Yin-tu

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Matemàtics --- Història --- Ramanujan Aiyangar, Srinivasa, --- Índia --- Ciències socials --- Humanitats --- Ciències polítiques --- Conspiracions --- Cops d'Estat --- Cròniques --- Cronologia --- Demografia històrica --- Descobriments geogràfics --- Diplomàcia --- Escàndols --- Filosofia de la història --- Fonts històriques --- Història antiga --- Història constitucional --- Història contemporània --- Història de l'antropologia --- Història de l'art --- Història de l'urbanisme --- Història de la ciència --- Història de la civilització --- Història de la dona --- Història de la filosofia --- Història de la lingüística --- Història de la literatura --- Història de la psicologia --- Història de la tecnologia --- Història de la veterinària --- Història de les biblioteques --- Història del dret --- Història del llibre --- Història del transport --- Història eclesiàstica --- Història econòmica --- Història local --- Història medieval --- Història militar --- Història moderna --- Història pública --- Història social --- Història universal --- Historiografia --- Llocs històrics --- Migració de pobles --- Reis i sobirans --- Cinema històric --- Protohistòria --- Sociologia històrica --- Ciències auxiliars de la història --- Didàctica de la història --- Historiadores --- Historiadors --- Previsió --- Científics --- Dones matemàtiques --- Matemàtica --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanujan, S. --- Ramanujam, S. --- Ramanujan, Srinivasa, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanujan Aiyangar, Srinivasa --- Bharat --- República de la India --- República de l'Índia --- Àsia del Sud --- Ajantha (Índia) --- Assam (Índia : Estat) --- Calcuta (Índia) --- Delhi (Índia) --- Kodagu (Índia : Regió) --- Goa (Índia : Estat) --- Jammu i Caixmir (Índia : Estat) --- Kerala (Índia : Estat) --- Madràs (Índia) --- Madhya Pradesh (Índia : Estat) --- Maharashtra (Índia : Estat) --- Orissa (Índia : Estat) --- Panjab (Índia : Estat) --- Rajasthan (Índia : Estat) --- Uttar Pradesh (Índia : Estat) --- Caixmir (Àsia : Regió) --- Mathematics. --- Math --- Science --- Brahmaputra (Àsia : Curs d'aigua)