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An Invitation to Modern Number Theory Hardcover – March 26, 2006
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"This is a great book. . . . [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."--Henry Ricardo, MAA Reviews
"This is the first text to present Random Matrix Theory and the Circle Method for German primes. This well-written book supplements classic texts by showing connections between seemingly diverse topics, by making the subject accessible to beginning students and by whetting their appetite for continuing in mathematics"--Mathematical Reviews
"I would highly recommend this book to anybody interested in number theory, from an undergraduate student to an established expert, since everybody will be able to find in this book lots of new interesting material, tempting problems, and interesting computational challenges. It could also be used as a textbook for a graduate course in number theory. To promote and stimulate independent research, it contains many very interesting exercises and even suggestions for research projects."--Igor Shparlinski, SIAM Review
From the Inside Flap
"The book provides a much-needed introduction to modern number theory that emphasizes analytic number theory. It should serve remarkably well as an advanced undergraduate textbook and its latter parts would be suitable for a beginning graduate course. Some of the material covered, such as the circle method and random matrix theory, is not readily available elsewhere in book form. These topics provide terrific examples of areas in number theory of great current interest that can be penetrated by students. I would seriously consider using this book in my own classes and recommend it with enthusiasm for highly motivated students."--William Duke, University of California, Los Angeles
"Having this selection of material available in essentially self-contained form is fantastic. Reading the book (or taking a class based on it) might easily decide the future endeavors of many a neophyte mathematician. I have yet to discover a clearer exposition of the works of the circle method. The inclusion of exercises and, especially, of problems for further research and theoretical or numerical exploration is extremely valuable. I would dare to compare the book to Hardy and Wright's classic An Introduction to the Theory of Numbers in that Miller and Takloo-Bighash expose readers to the lively work of number theory, to its proofs, ideas, and methods, assuming only a very modest background."--Eduardo Dueñez, University of Texas, San Antonio
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As for the book, if you have read any of the Burtons(my actual favorite) or perhaps a silverman there is a general outline. This book doesn't follow those exact lines but I think it is a better book in some respects. In some ways a small step ahead of them as perhaps the writer intended to skip some of the stuff that is in every introductory number theory book. If you like this then don't overlook apostol as your next number theory book. enjoy
except those already familiar with the material-- why the
authors opted for this broad and sketchy treatment is unclear ---
introducing more topics than a russian novel has characters,
the book is more likely to lead the reader to a state of exhaustion
rather than enlightenment -- one has to ask who could possibly benefit
from this machine gun style exposition as there are more comprehensive
and gentle surveys available of each of the many subjects they touch on --
answer = no one -- ps it is an abomination to compare this unwieldy mess
to a pedagogical masterpiece like Hardy and Wright -- don't believe their
The exposition is brilliant, rigorous, well paced, absolutely non-flippant and elegant (it feels like I am reading a latexed version of a G.H. Hardy book).
It is highly innovative since it has material that you normally do not expect to find in one single book. There are books about probabilistic number theory, but those books are devoted wholly to that subject. Same thing goes for random matrix theory. But the most surprising case is that of continued fractions. Books on continued fractions are *generally* elementary and not very long.
No single book out there combines introductions to the interactions of probability and random matrix theory with number theory in addition to treating more standard subjects (cryptography, group theory, continued fractions, circle method, L-functions, ...) exquisitely. This has been wonderfully achieved by Miller and Takloo-Bighash.
All in all, the flavour of this book is best summarized with the word: modern.
This book is not a popular math book. Yet not quite a textbook either, it is, as the title suggests, an invitation. And a serious invitation, for that matter. A little effort will be needed, however I have found out that whatever amount of effort you invest in it will be rewarded with interests!
I would say that courses in group theory, elementary number theory and complex analysis would constitute an adequate background.
It need not be read linearly, which is also a bonus. Within reason, you could move on to whichever subjects you find more interesting.
(By the way, the introductions to Fourier analysis and probability theory put certain 'methods' books to shame!)
My favourite parts are the summary of the status of the Riemann Hypothesis. It made me interested in finding out more about the Xian-Jin Li's criterion, Hardy's Theorem, the whole function-theoretic aspect of the book, and the connection between number theory and physics.
Finally, financially speaking, it is *very* well priced for a hardcover.