Ramanujan's sum

About: Ramanujan's sum is a research topic. Over the lifetime, 3663 publications have been published within this topic receiving 49932 citations.

The lost notebook and other unpublished papers

Abstract: The so-called "Lost Notebook" of S.R. Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens. A major portion of the "Lost Notebook" - really just 90 unpaginated sheets of work on "q"-series and other topics - is reproduced here in facsimile. Letters from Ramanujan to Hardy as well as various other sheets of seemingly related notes are then included, on topics including coefficients in the 1/q3 and 1/q2 problems and the mock theta functions. The next 180 pages consist of unpublished manuscripts of Ramanujan, including 28 pages from the 'Loose Papers held in the Trinity College Library. Finally a number of interesting letters that were exchanged between Ramanujan, Littlewood, Hardy and Watson, with a bearing on Ramanujan's work are collected together here with other extracts and fragments.

Ramanujan's Notebooks

Abstract: Working mostly in isolation, Ramanujan noted striking and sometimes still unproved results in series, special functions and number theory.

Discrete Groups, Expanding Graphs and Invariant Measures

Abstract: Expanding Graphs.- The Banach-Ruziewicz Problem.- Kazhdan Property (T) and its Applications.- The Laplacian and its Eigenvalues.- The Representation Theory of PGL 2.- Spectral Decomposition of L 2(G(?)\G(A)).- Banach-Ruziewicz Problem for n = 2, 3 Ramanujan Graphs.- Some More Discrete Mathematics.- Distributing Points on the Sphere.- Open Problems.

Ramanujan's Lost Notebook: Part II

Abstract: 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 thefourthof fivevolumes thatthe authors plan to write on Ramanujans lost notebook.In contrast to thefirst three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems.Reviewfrom 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."- MathSciNetReview 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