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Title: Multiplicative Number Theory by Harold Davenport, H.L. Montgomery, S. Axler, F.W. Gehring, K.A. Ribet, Hugh L. Montgomery ISBN: 0-387-95097-4 Publisher: Springer-Verlag Pub. Date: 15 January, 2000 Format: Hardcover Volumes: 1 List Price(USD): $54.95 |
Average Customer Rating: 5 (2 reviews)
Rating: 5
Summary: An extraordinary Book
Comment: Ever since I first read about the prime number theorem, I have been roaming the mathetmatical landscape, looking for the best proof of this result. I believe this book has it. It's not the simplest or the shortest proof, but it gives the deepest understanding of why the prime numbers behve like they do. In addition to this, it shows you the historical perspective in these proofs. All too often today math books give one short and slick proofs that leave you wondering how on earth they came up with it. In this book, however, one can almost feel the thoughts going through Riemann and Dirichlet's heads as they came up with the theorems. This book also has the proof of Dirichlet's theorem and Vinogradov's partial proof of the ternary goldbach conjecture. The vinogradov and following sections are considerably harder, partly because they were not written by Davenport himself. Anyway, if you're serious about Analytic number theory and how mathematicians think, this books needs to be on your bookshelf.
Rating: 5
Summary: A good historical approach to Analytic Number Theory
Comment: I like this book because it gives you a good understanding of where the difficulties in the subject are. It takes a historical approach, following more or less the same steps that the original discoverers of these results took. Today we have very slick proofs for many of these results, and it is sometimes hard to understand why it took so long to discover them in the first place, but this book will give you this understanding; Dirichlet in particular practically had to invent Analytic Number Theory to prove his theorem on primes in an arithmetic progression.
The book works up gradually to each result, for example proving Dirichlet's theorem first for a prime modulus (as Dirichlet did himself), then the general modulus. In most cases it proves first the result for all primes (zeta function) and then the generalization for primes in an arithmetic progression (L function), pointing out which parts generalize easily and which cause special difficulties.
Some of the more advanced results covered are exponential sums, Vinogradov's theorem that every large odd number is the sum of three primes, and Bombieri's theorem about the average distribution of primes in arithmetic progressions.
I haven't seen the previous (1980) edition; this new edition seems to be lightly revised from the previous one. The last chapter is up-to-date and gives a brief survey of new results and of new books on the subject.
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Title: Riemann's Zeta Function by Harold M. Edwards ISBN: 0486417409 Publisher: Dover Publications Pub. Date: 01 June, 2001 List Price(USD): $14.95 |
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Title: The Distribution of Prime Numbers (Cambridge Mathematical Library) by A.E. Ingham ISBN: 0521397898 Publisher: Cambridge University Press Pub. Date: 01 November, 1990 List Price(USD): $19.00 |
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Title: An Introduction to the Langlands Program by Daniel Bump, Stephen Gelbart, Joseph Bernstein ISBN: 0817632115 Publisher: Birkhauser Boston Pub. Date: June, 2003 List Price(USD): $39.95 |
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Title: The Riemann Zeta-Function: Theory and Applications by A. Ivic ISBN: 0486428133 Publisher: Dover Publications Pub. Date: 01 June, 2003 List Price(USD): $29.95 |
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Title: Continued Fractions by Aleksandr Iakovlevich Khinchin, A. Ya. Khinchin, Herbert Eagle ISBN: 0486696308 Publisher: Dover Publications Pub. Date: 01 May, 1997 List Price(USD): $8.95 |
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