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Title: Riemann's Zeta Function by Harold M. Edwards ISBN: 0-486-41740-9 Publisher: Dover Pubns Pub. Date: 13 June, 2001 Format: Paperback Volumes: 1 List Price(USD): $14.95 |
Average Customer Rating: 5 (4 reviews)
Rating: 5
Summary: Good complement to Ivic and Titchmarsh
Comment: This is by far the book of mathematics that I like most. It's not the most complete source of information about the zeta function, Titchmarsh and Ivic are the authorities. However when you read this book, you have a feeling that you are following Riemann's, de la Vallée Poussin's, Hadamard's, Littlewood's, etc... steps and you understand how these mathematicians must have felt while they studied the zeta function.
It includes a translation of Riemann's original paper (On the Number of Primes...) which is very nice and most authors now seem to forget to mention (mainly because of the obscure way in which it was written).
The first chapter is devoted to the study of the paper, then it is followed another chapter proving the product formula (which was not quite proven by Riemann), then a third chapter of von Mangoldt's proof of Riemann's Prime Formula.
The fourth chapter has the famous prime number theorem and it's original proof by Hadamard and Poussin. The fifth one includes an error estimation due to Poussin for the prime number theorem, and the equivalent of the Riemann Hypothesis in terms of prime distributions.
The Euler-Maclaurin formula is introduced in the sixth chapter to calculate zeros in the critical line.
The Riemann-Siegel formula is introduced in the seventh, and then later chapters include large scale computations, Fourier analysis, growth and location of zeros.
Finally we have my favourite chapter, counting zeros: Hardy's theorem, which says that there are infinitely many zeros in the critical line, which was improved by Littlewood, then later by Selberg, and then by Levinson.
The last chapter is dedicated to some theorems, including an elementary proof of the prime number theorem.
Most important idea: the introduction! It will give you an idea of how these amazing people studied and did math.
Rating: 5
Summary: New and old.
Comment: The popular press leaves us with the impression that math is
intimidating. This wasn't always the case. In my time, the approach to how we teach math went thru cycles: (1) The boot-camp
approach with its endless drills, (2) The New-Math approach, (3) The back-to-basics trend, and (4) The Make-it-Seem-Easy-and Fun approach and the motivational speakers.---Finally Edwards suggests, following Eric Temple Bell, that we rather begin with the classics when approaching a subject in math. It was thought that later books based on the classics had more effective ways of doing it, and few took the trouble of looking at the original and central papers of the great masters. The landmark papers. All the while, they collected dust on the shelves in the back rooms of libraries. Of the classics, the true landmarks, one stands out: It is Riemann's paper on the prime numbers, what later turned into the prime number theorem. It is also the paper with the Riemann hypothesis, still unproved, now generations later. So it is a delightful idea including Riemann's paper, in translation, in an appendix. It would have been nice had Edwards also reproduced the original German text. Now the RH is one of the Million-Dollar problems in math. It is anyone's guess when it will be cracked, but in the mean time, it continues to inspire generations of mathematicians and students. This Dover edition is came out in 2001. The original first 1974 edition, Academic Press, had gone out of print. This lovely book seems still to be a model that we can measure other books against. Edwards' presentation is both engaging and deep, and the book contains the gems in a subject that continues to be central in math, the subject of analytic number theory.
Rating: 5
Summary: A must for all who are interested in Riemann Zeta fuction!
Comment: I have both bought this book and Titchmarsh's one. Both are classics of that subject. Titchmarsh's one is more difficult to read though is even more comphrehensive!. Edward's one is more concise and is more easy to read === One specific point about this book whereas all other books do not have is that it includes the original paper ( in translation) of Riemann's original classic paper. I think this is very important and was neglected by all other books on this subject. From that not only we can have a more thorough understanding to what Riemann originally thinked and developed his famous function and this also serves as a respect to Riemann, one of the three greatest mathematicians of modern times!! ( the other two being Euler and Gauss. Newton, the greatest one of them all was not included as we usually do not include him in these periods)
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Title: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire ISBN: 0309085497 Publisher: Joseph Henry Press Pub. Date: 23 April, 2003 List Price(USD): $27.95 |
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Title: Gamma : Exploring Euler's Constant by Julian Havil ISBN: 0691099839 Publisher: Princeton Univ Pr Pub. Date: 17 March, 2003 List Price(USD): $29.95 |
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Title: Three Pearls of Number Theory by A. Y. Khinchin ISBN: 0486400263 Publisher: Dover Pubns Pub. Date: 18 February, 1998 List Price(USD): $6.95 |
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Title: Continued Fractions by A. Ya. Khinchin ISBN: 0486696308 Publisher: Dover Pubns Pub. Date: 14 May, 1997 List Price(USD): $8.95 |
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Title: Number Theory by George E. Andrews ISBN: 0486682528 Publisher: Dover Pubns Pub. Date: 12 October, 1994 List Price(USD): $8.95 |
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