- Paperback: 656 pages
- Publisher: Oxford University Press; 6 edition (September 15, 2008)
- Language: English
- ISBN-10: 0199219869
- ISBN-13: 978-0199219865
- Product Dimensions: 9.1 x 1.3 x 6.1 inches
- Shipping Weight: 2.2 pounds (View shipping rates and policies)
- Average Customer Review: 22 customer reviews
- Amazon Best Sellers Rank: #96,384 in Books (See Top 100 in Books)
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An Introduction to the Theory of Numbers 6th Edition
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`Review from previous edition Mathematicians of all kinds will find the book pleasant and stimulating reading, and even experts on the theory of numbers will find that the authors have something new to say on many of the topics they have selected... Each chapter is a model of clear exposition,
and the notes at the ends of the chapters, with the references and suggestions for further reading, are invaluable.'
`This fascinating book... gives a full, vivid and exciting account of its subject, as far as this can be done without using too much advanced theory.'
`...an important reference work... which is certain to continue its long and successful life...'
`...remains invaluable as a first course on the subject, and as a source of food for thought for anyone wishing to strike out on his own.'
About the Author
Roger Heath-Brown F.R.S. was born in 1952, and is currently Professor of
Pure Mathematics at Oxford University. He works in analytic number
theory, and in particular on its applications to prime numbers and to
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Top customer reviews
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Each chapter stands mostly by itself, so for example one doesn't need to have read the earlier chapters to understand the chapters on generating functions and orders of magnitude of arithmetical functions and on the partition function, which were the first chapters I first as an undergraduate. The chapters on continued fractions and Diophantine approximation are good, and also don't depend on other parts of the book. I have that feeling that continued fractions are generally seen as (i) a curiosity, (ii) an odd tool to prove counterexamples in analysis (like the Cantor set), or (iii) as a specialized area, that perhaps descriptive set theorists or people working in transcendental number theory care about. In fact, continued fractions ought to be a part of the analyst's arsenal: they are the most natural way of representing real numbers by sequences of integers, and appear often in measure theory, especially when trying to construct explicit examples, like Lusin's set that is not Borel measurable (but which is Lebesgue measurable). Several proofs of Kronecker's theorem are given, which is a result that is probably more important in dynamical systems than it is in number theory by itself. There is a chapter on the geometry of numbers in which Minkowski's theorem, about symmetric convex sets of great enough volume containing lattice points, is proved. This book is probably the best place to read Selberg's proof of the prime number theorem. After praising the book, let me say that I have never been attracted to the chapters on quadratic fields, and of the whole book I think these are the most dated, and one might better use the sections in Dummit and Foote Abstract Algebra, 3rd Edition or for a serious presentation, Swinnerton-Dyer A Brief Guide to Algebraic Number Theory (London Mathematical Society Student Texts), or for a marvellous book just on quadratic fields that connects quadratic fields to Gauss's work on quadratic forms and later work on modular forms, Cox Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication.
I recommend this book for any mathematically trained reader. It doesn't have to be read cover to cover, and you will enjoy opening it for years. It is too broad to be used for a course and has no exercises, but for the independent reader, I recommend reading the chapter you choose from start to finish. If the chapter refers to statements proved in earlier chapters, you should carefully read the statement that is being used, understand what it says, but don't spend a minute trying to prove it, because that might send you even farther back checking other statements. You should carefully read all the proofs in the chapter and all the examples and make sure you understand every step, not just agree that it seems like a thing that could be correct but that you are sure why it is true. There may be parts that don't make sense, and if you don't have any help then try reading another chapter and coming back later.
This book is an example of how much good math can be explained without on the one hand demanding any prior machinery and without on the other hand describing math rather than doing math (like how every educated person has heard about Schrödinger's cat, and the shallow things they have heard probably leave them knowing less than if they had heard nothing at all, yet like food that is not nutritious, these shallow explanations fill the stomach and let you think you know something), and the closest comparison I can think of is Hilbert and Cohn-Vossen Geometry and the Imagination (AMS Chelsea Publishing).