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Edwards elaborates on Bernard Riemann's eight-page paper On the Number of Primes Less Than a Given Magnitude, published in German in 1859. His goal is not to supplant the classic work, but to provide mathematics students access to it. Indeed an English translation of the original is appended. Academic Press published the 1974 edition. Cited in Book
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This book is a study of Bernhard Riemann's epoch-making 8-page paper "On the Number of Primes Less Than a Given Magnitude," and of the subsequent developments in the theory which this paper inaugurated. Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
This book is a study of Bernhard Riemann's epoch-making 8-page paper "On the Number of Primes Less Than a Given Magnitude," and of the subsequent developments in the theory which this paper inaugurated. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Customer Reviews
Most Helpful Customer Reviews
57 of 68 people found the following review helpful
Format:Paperback
The popular press leaves us with the impression that math is
intimmidating. 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 lovely 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.
intimmidating. 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 lovely 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.
Format:Paperback
Probably the best book to explain the Zeta function and the problem of proving it. Needs some advanced maths knowledge to get the best out of it, so not for the layman. Like all the other books on the subject that I have read, suffers from not giving a clear and detailed description of calculating a numeric example. This book is recommended in other books on the subject, so if you are interested buy this one first.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
16 reviews
86 of 87 people found the following review helpful
Good complement to Ivic and Titchmarsh
19 Jan 2004
Format:Paperback
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.
39 of 39 people found the following review helpful
This book is great
13 July 2005
Format:Paperback
It has always seemed to me that the very best modern books on the Riemann Zeta Function, and its applications to analytic number theory, are either written at a vey high or a very low level of mathematical sophistication. This book successfully bridges the gap between the uninformative "popular texts" and extremely advanced texts on analytic NT. True, you won't find material on generalized Dirichlet L-Functions, modular forms, advanced spectral theory of self-adjoint operators, and other such things in this book, nor will you find hopelessly obscurely worded, nonrigorous explanations like in "popular" math books; what you will find is an exposition of all the most important aspects of the theory which is accessible to anyone with even a piecemeal knowledge of real analysis and the rudiments of the theory of series and integrals of functions of a complex variable. The statement on the back cover that the "mathematically inclined general reader" will find this book accessible is certainly untrue when it comes to most such readers, but I would recommend this book to anyone with a basic knowledge of analysis and number theory who wants to really understand the math behind this important subject without overextending himself mathematically.
46 of 48 people found the following review helpful
Excellent for experts and the casual mathematician alike
19 Sep 2005
Format:Paperback|Amazon Verified Purchase
I hesitate to add to the chorus of praise here for H.M. Edwards's "Riemann's Zeta Function," for what little mathematics I have is self taught. Nevertheless, after reading John Derbyshire's gripping "Prime Obsession" and following the math he used there with ease, I thought to tackle a more challenging book on the subject. A Topologist friend suggested Titchmarsh's "The Theory of the Riemann Zeta-Function," but I soon bogged down. I happily came across Edwards while browsing, and was pleased both with the low price, and the lucid contents.
For those who are mathematicians and like their introductions to the most fascinating math problems straight and touching all horizons of inquiry, then experts appear to have converged on Titchmarsh as the volume for the first string. However, Edward's work is also appropriate for experts and hits the highlights of background leading to the Zeta function. But Edward's chief strength is beyond his intended audience, for it is his accessibility for the occasional mathematician. With some patience, and not without some little pain and an occasional side trip to "The World of Mathematics" or "The Encyclopedia of Mathematics," even a self-trained mathematician can appreciate most of what Edwards is explaining.
In short, I heartily recommend to those who have enjoyed John Derbyshire's "Prime Obsession," and have additional steam, to take up Edward's "Riemann' Zeta Function" volume for further insights and knowledge.
For those who are mathematicians and like their introductions to the most fascinating math problems straight and touching all horizons of inquiry, then experts appear to have converged on Titchmarsh as the volume for the first string. However, Edward's work is also appropriate for experts and hits the highlights of background leading to the Zeta function. But Edward's chief strength is beyond his intended audience, for it is his accessibility for the occasional mathematician. With some patience, and not without some little pain and an occasional side trip to "The World of Mathematics" or "The Encyclopedia of Mathematics," even a self-trained mathematician can appreciate most of what Edwards is explaining.
In short, I heartily recommend to those who have enjoyed John Derbyshire's "Prime Obsession," and have additional steam, to take up Edward's "Riemann' Zeta Function" volume for further insights and knowledge.
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