Bruce C. Berndt

Harold Diamond

Adolf J Hildebrand

The second of two volumes presenting papers from an international conference on analytic number theory. The two volumes contain 50 papers, with an emphasis on topics such as sieves, related combinatorial aspects, multiplicative number theory, additive number theory, and Riemann zeta-function.

Analytic Number Theory:The Halberstam Festschrift 2

Ken Ono

Robert A. Rankin, one of the world's foremost authorities on modular forms and a founding editor of The Ramanujan Journal, died on January 27, 2001, at the age of 85. Rankin had broad interests and contributed fundamental papers in a wide variety of areas within number theory, geometry, analysis, and algebra. To commemorate Rankin's life and work, the editors have collected together 25 papers by several eminent mathematicians reflecting Rankin's extensive range of interests within number theory. Many of these papers reflect Rankin's primary focus in modular forms. It is the editors' fervent hope that mathematicians will be stimulated by these papers and gain a greater appreciation for Rankin's contributions to mathematics. This volume would be an inspiration to students and researchers in the areas of number theory and modular forms.

Papers in Memory of Robert A. Rankin

Number Theory and Modular Forms

George E. Andrews

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plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988.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

Part III

Ramanujan's Lost Notebook

The fifth and final volume to establish the results claimed by the great Indian mathematician Srinivasa Ramanujan in his "Notebooks" first published in 1957. Although each of the five volumes contains many deep results, the average depth in this volume is possibly greater than in the first four. There are several results on continued fractions - a subject that Ramanujan loved very much. It is the authors wish that this and previous volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujans remarkable ideas.

Part V

Ramanujan’s Notebooks

Krishnaswami Alladi

Peter Paule

Ae Ja Yee

James A. Sellers

This book presents a printed testimony for the fact that George Andrews, one of the world’s leading experts in partitions and q-series for the last several decades, has passed the milestone age of 80. To honor George Andrews on this occasion, the conference “Combinatory Analysis 2018” was organized at the Pennsylvania State University from June 21 to 24, 2018. This volume comprises the original articles from the Special Issue “Combinatory Analysis 2018 – In Honor of George Andrews’ 80th Birthday” resulting from the conference and published in Annals of Combinatorics. In addition to the 37 articles of the Andrews 80 Special Issue, the book includes two new papers. These research contributions explore new grounds and present new achievements, research trends, and problems in the area. The volume is complemented by three special personal contributions: “The Worlds of George Andrews, a daughter’s take” by Amy Alznauer, “My association and collaboration with George Andrews” by Krishna Alladi, and “Ramanujan, his Lost Notebook, its importance” by Bruce Berndt. Another aspect which gives this Andrews volume a truly unique character is the “Photos” collection. In addition to pictures taken at “Combinatory Analysis 2018”, the editors selected a variety of photos, many of them not available elsewhere: “Andrews in Austria”, “Andrews in China”, “Andrews in Florida”, “Andrews in Illinois”, and “Andrews in India”. This volume will be of interest to researchers, PhD students, and interested practitioners working in the area of Combinatory Analysis, q-Series, and related fields.

George E. Andrews 80 Years of Combinatory Analysis

On May 16 -20, 1995, approximately 150 mathematicians gathered at the Conference Center of the University of Illinois at Allerton Park for an Inter national Conference on Analytic Number Theory. The meeting marked the approaching official retirement of Heini Halberstam from the mathematics fac ulty of the University of Illinois at Urbana-Champaign. Professor Halberstam has been at the University since 1980, for 8 years as head of the Department of Mathematics, and has been a leading researcher and teacher in number theory for over forty years. The program included invited one hour lectures by G. Andrews, J. Bour gain, J. M. Deshouillers, H. Halberstam, D. R. Heath-Brown, H. Iwaniec, H. L. Montgomery, R. Murty, C. Pomerance, and R. C. Vaughan, and almost one hundred other talks of varying lengths. These volumes comprise contributions from most of the principal speakers and from many of the other participants, as well as some papers from mathematicians who were unable to attend. The contents span a broad range of themes from contemporary number theory, with the majority having an analytic flavor.

Proceedings of a Conference In Honor of Heini Halberstam

Analytic Number Theory

Harold G. Diamond

Adolf J. Hildebrand

Proceedings of a Conference In Honor of Heini Halberstam Volume 1

Upon Ramanujans death in 1920, G. H. Hardy strongly urged that Ramanujans notebooks be published and edited. In 1957, the Tata Institute of Fundamental Research in Bombay finally published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the task of editing Ramanujans notebooks: proofs are provided to theorems not yet proven in previous literature, and many results are so startling as to be unique.

K. Ramachandra

Michel Waldschmidt

Christian F. Krattenthaler

A.K. Agarwal

Gary L. Mullen

To mark the World Mathematical Year 2000 an International Conference on Number Theory and Discrete Mathematics in honour of the legendary Indian Mathematician Srinivasa Ramanuj~ was held at the centre for Advanced study in Mathematics, Panjab University, Chandigarh, India during October 2-6, 2000. This volume contains the proceedings of that conference. In all there were 82 participants including 14 overseas participants from Austria, France, Hungary, Italy, Japan, Korea, Singapore and the USA. The conference was inaugurated by Prof. K. N. Pathak, Hon. Vice-Chancellor, Panjab University, Chandigarh on October 2, 2000. Prof. Bruce C. Berndt of the University of Illinois, Urbana Chaimpaign, USA delivered the key note address entitled "The Life, Notebooks and Mathematical Contributions of Srinivasa Ramanujan". He described Ramanujan--as one of this century's most influential Mathematicians. Quoting Mark K. ac, Prof. George E. Andrews of the Pennsylvania State University, USA, in his message for the conference, described Ramanujan as a "magical genius". During the 5-day deliberations invited speakers gave talks on various topics in number theory and discrete mathematics. We mention here a few of them just as a sampling: • M. Waldschmidt, in his article, provides a very nice introduction to the topic of multiple poly logarithms and their special values. • C.

Number Theory and Discrete Mathematics

During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

Srinivasa Ramanujan Aiyangar

Robert Alexander Rankin

This book contains essays on Ramanujan and his work that were written especially for this volume. It also includes important survey articles in areas influenced by Ramanujan's mathematics. Most of the articles in the book are nontechnical, but even those that are more technical contain substantial sections that will engage the general reader.

Essays and Surveys

Ramanujan: Essays and Surveys

Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of $q$-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics.The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

Number Theory in the Spirit of Ramanujan

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

Bruce (University of Illinois Berndt

The subject of $q$-series can be said to begin with Euler and his pentagonal number theorem. In fact, $q$-series are sometimes called Eulerian series. Contributions were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic development, especially from the point of view of studying series with the products in the summands, was made by E. Heine in 1847. In the latter part of the nineteenth and in the early part of the twentieth centuries, two English mathematicians, L. J. Rogers and F. H. Jackson, made fundamental contributions.In 1940, G. H. Hardy described what we now call Ramanujan's famous $_1\psi_1$ summation theorem as 'a remarkable formula with many parameters'. This is now one of the fundamental theorems of the subject. Despite humble beginnings, the subject of $q$-series has flourished in the past three decades, particularly with its applications to combinatorics, number theory, and physics. During the year 2000, the University of Illinois embraced The Millennial Year in Number Theory. One of the events that year was the conference $q$-Series with Applications to Combinatorics, Number Theory, and Physics. This event gathered mathematicians from the world over to lecture and discuss their research. This volume presents nineteen of the papers presented at the conference. The excellent lectures that are included chart pathways into the future and survey the numerous applications of $q$-series to combinatorics, number theory, and physics.

Bruce C. Berndt

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