Number Theory in the Spirit of Ramanujan by Bruce C. Berndt (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “… undergraduates will find no better place to meet the mind behind the towering reputation.” —- D. V. Feldman, University of New Hampshire for CHOICE Reviews”This is a delightful little book on selected topics in number theory. …I highly recommend this book to all mathematicians. It is a great resource both to learn from and to teach from. Even the experts will enjoy his new perspective on these old questions.” —- Journal of Approximation Theory”This slender volume is extremely well-written and contains a wealth of material. It is a lucid and accessible introduction to a rich and fascinating area of mathematics, written by the world’s leading expert. For anyone with a knowledge of calculus wanting to learn about the mathematical work of Ramanujan, this book is the best place to start.” —- Shaun Cooper, Massey University – New Zealand Newsletter

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Requires a high level of number theory and advanced mathematics

⭐Product quantity is good and content is also good. After reading this book I’m giving a details.

⭐Super

⭐インドが生んだ天才ラマヌジャンの数論上の発見を証明付きで解説した書物は少なく、初学者向けの解説書となると、『数論II』（岩波）の第9章（「保型形式とは」）やハーディの『Ramanujan』など、数える程しかない。本書はラマヌジャンのノートブック研究の第一人者であるバーント教授による待望の入門書である。ラマヌジャンの数論上の多くの発見が、q級数や彼独自のテータ関数論、更にアイゼンシュタイン級数を含むランベルト級数の巧妙な数式変形によって導かれている事を本書によって知ることができる。正則アイゼンシュタイン級数に関する2つの素晴らしい漸化式を述べた4.2節は本書の一つのハイライトであり、この解説を読まれた方は「これは凄い！」と感嘆の声をあげられると思う。超幾何関数（特に、第一種完全楕円積分）とラマヌジャンのテータ関数の関係を述べた第5章も抜群に面白い。ラマヌジャンのテータ関数（φ、ψ、f、χ）とアイゼンシュタイン級数（P、Q、R）を楕円パラメータ（x、z）で明示的に表示する5.4節の公式群、及びその応用として平方数の和による自然数の表現公式やモジュラー方程式を導くという方法を私はこの本で初めて教えられた。本書を読めば、ラマヌジャンが発見した多くの美しい恒等式を見出す事が出来る筈である。数学的な「事実」、「証明」、及びその事実を発見するに至った「思考過程」のTriadについて、考えさせられる書でもあると思う。この面ではハーディの好著『Ramanujan』がとても参考になるので、併せて一読される事をお薦めしたい。Its awsome….genius ..ramanujan …

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