Description : Originally published in 1927, this book presents the collected papers of the renowned Indian mathematician Srinivasa Ramanujan (1887-1920), with editorial contributions from G. H. Hardy (1877-1947). Detailed notes are incorporated throughout and appendices are also included. This book will be of value to anyone with an interest in the works of Ramanujan and the history of mathematics.
Description : These two volumes contain all the papers published by Hans Rademacher, either alone or as joint author, essentially in chronological order. Included also are a collection of published abstracts, a number of papers that appeared in institutes and seminars but are only now being formally published, and several problems posed and/or solved by Rademacher. The editor has provided notes for each paper, offering comments and making corrections. He has also contributed a biographical sketch. The earlier papers are on real variables, measurability, convergence factors, and Euler summability of series. This phase of Rademacher's work culminates in a paper of 1922, in which he introduced the systems of orthogonal functions now known as the Rademacher functions. After this, a new period in Rademacher's career began, and his major effort was devoted to the theory of functions of a complex variable and number theory. Some of his most important contributions were made in these fields. He perfected the sieve method and used it skillfully in the study of algebraic number fields; he studied the additive prime number theory of these fields; he generalized Goldbach's Problem; and he began his work on the theory of the Riemann zeta function, modular functions, and Dedekind sums (now often&-and justly&-called Dedekind-Rademacher sums). To this period also becomes what has become known as the Rademacher-Brauer formula. Rademacher came to the United States as a refugee in 1934. In the years that followed, he obtained some of his most important results in connection with the Fourier coefficients of modular forms of positive dimensions. His general method may be considered a modification and improvement of the Hardy-Ramanujan-Littlewood circle method. He also published additional papers on Dedekind-Rademacher sums (with A. Whiteman), general number theory (with H. S. Zuckerman), and modular functions (also with Zuckerman). During the last decade of his life&-the 1960s&-he continued his work on these problems and devoted considerable attention to general analysis&-especially harmonic analysis&-and to analytic number theory. All of the papers in Volume I and ten of those in Volume II are in German. One paper is in Hungarian. The volumes are part of the MIT Press series Mathematicians of Our Time (Gian-Carlo Rota, general editor).
Description : Vijay Kumar Patodi was a brilliant Indian mathematicians who made, during his short life, fundamental contributions to the analytic proof of the index theorem and to the study of differential geometric invariants of manifolds. This set of collected papers edited by Prof M Atiyah and Prof Narasimhan includes his path-breaking papers on the McKean-Singer conjecture and the analytic proof of Riemann-Roch-Hirzebruch theorem for Kähler manifolds. It also contains his celebrated joint papers on the index theorem and the Atiyah-Patodi-Singer invariant.
Description : Number theory is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved, but more arise. All the research for these ancient or new problems implicated and are still promoting the development of number theory and mathematics.American-Romanian number theorist Florentin Smarandache introduced hundreds of interest sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1991, he published a book named Only problems, Not solutions!. He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it. Already many researchers studied these sequences and functions from his book, and obtained important results.This book, Research on Smarandache Problems in Number Theory (Collected papers), contains 41 research papers involving the Smarandache sequences, functions, or problems and conjectures on them.All these papers are original. Some of them treat the mean value or hybrid mean value of Smarandache type functions, like the famous Smarandache function, Smarandache ceil function, or Smarandache primitive function. Others treat the mean value of some famous number theoretic functions acting on the Smarandache sequences, like k-th root sequence, k-th complement sequence, or factorial part sequence, etc. There are papers that study the convergent property of some infinite series involving the Smarandache type sequences. Some of these sequences have been first investigated too. In addition, new sequences as additive complement sequences are first studied in several papers of this book.Most authors of these papers are my students. After this chance, I hope they will be more interested in the mysterious integer and number theory!More future papers by my students will focus on the Smarandache notions, such as sequences, functions, constants, numbers, continued fractions, infinite products, series, etc. in number theory!List of the Contributors:Zhang Wenpeng, Xu Zhefeng, Zhang Xiaobeng, Zhu Minhui, Gao Nan, Guo Jinbao, He Yanfeng, Yang Mingshun, Li Chao, Gao Jing, Yi Yuan, Wang Xiaoying, Lv Chuan, Yao Weili, Gou Su, He Xiaolin, Li Hailong, Liu Duansen, Li Junzhuang, Liu Huaning, Zhang Tianping, Ding Liping, Li Jie, Lou Yuanbing, Zhao Xiqing, Zhao Xiaopeng, Yang Cundian, Liang Fangchi