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Riemann's Zeta Function (Dover Books on Mathematics) [Englisch] [Taschenbuch]

Harold M. Edwards , Mickey Edwards , H. M. Edwards
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Produktinformation

  • Taschenbuch: 315 Seiten
  • Verlag: Dover Pubn Inc; Auflage: Dover. (28. März 2003)
  • Sprache: Englisch
  • ISBN-10: 0486417409
  • ISBN-13: 978-0486417400
  • Größe und/oder Gewicht: 21,6 x 13,8 x 1,8 cm
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen  Alle Rezensionen anzeigen (3 Kundenrezensionen)
  • Amazon Bestseller-Rang: Nr. 32.838 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)

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Synopsis

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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Einleitungssatz
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. Lesen Sie die erste Seite
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Die hilfreichsten Kundenrezensionen
15 von 15 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen New and old. 4. April 2003
Format:Taschenbuch
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.
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6 von 6 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen This is where you should start your study 2. Januar 2010
Format:Taschenbuch
This book gives a detailed overview of the Riemann Zeta function. It covers both the theoretical as well as practical applications of the function. Recommended as a starter for your study of this wonderful function. It misses recent results (it has been written 35 years ago), but once you get through it your ready for what comes after.
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5.0 von 5 Sternen Ein Klassiker 26. Juni 2013
Von WinWen
Format:Taschenbuch|Verifizierter Kauf
Wer tiefer in die Geheimnisse der Riemannschen Vermutung - dem größten Preisrätsel der modernen Mathematik eindringen will, findet hier ein Standardwerk, welches nicht nur Riemanns richtungsweisende Veröffentlichung ausführlich erklärt, sondern auch viele weiterführenden Veröffentlichungen und Entdeckungen mitsamt deren Herleitungen, Argumentationen und Beweise verständlich darstellt.
Es ist ein Klassiker und -wenngleich mittlerweile zahlreiche andere Methoden entwickelt wurden, um dem Problem zu Leibe zu rücken- wird es noch lange ein Klassiker bleiben. Für den Weg zu einem tieferen Verständnis dieses Problems unverzichtbar!
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Amazon.com: 4.7 von 5 Sternen  21 Rezensionen
95 von 96 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Good complement to Ivic and Titchmarsh 19. Januar 2004
Von J. N. M. ROBLES - Veröffentlicht auf Amazon.com
Format:Taschenbuch
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.
46 von 47 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen This book is great 13. Juli 2005
Von MathGeek741 - Veröffentlicht auf Amazon.com
Format:Taschenbuch
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.
55 von 58 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Excellent for experts and the casual mathematician alike 19. September 2005
Von Bachelier - Veröffentlicht auf Amazon.com
Format:Taschenbuch|Verifizierter Kauf
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.
54 von 66 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen New and old. 4. April 2003
Von Palle E T Jorgensen - Veröffentlicht auf Amazon.com
Format:Taschenbuch|Verifizierter Kauf
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.
12 von 14 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen A good guide to Riemann, the prime number theorem, and the Riemann hypothesis 10. April 2006
Von Viktor Blasjo - Veröffentlicht auf Amazon.com
Format:Taschenbuch
Chapter 1 analyses Riemann's paper in detail. The zeta function is the product over all primes of 1/(1-1/p^s). Taking the logarithm, one obtains an expression involving the density of primes. So to say something about the density of primes one must say something about the log of the zeta function. Riemann does this by allowing the variable s of the zeta function to be complex, which enables him to prove the functional equation of the zeta function and the product representation of the xi function defined through it. From here he can derive an expression for log zeta, thus yielding an expression for prime density. Since it comes from log zeta, this expression depends on the poles of log zeta, i.e. the zeros of the zeta function. Riemann feels that all nontrivial zeros have real part 1/2, but this doesn't really matter right now since the term in the prime density expression depending on the zeros is "periodic" in any case and Riemann thus discards it without much harm when he derives his expression for the number of primes less than x. Hadamard and von Mangoldt later gave more rigourous proofs of the product formula for the xi function (chapter 2) and Riemann's prime density expression (chapter 3). As indicated by Riemann, further progress depends on an understanding of the zeros of the zeta function. Indeed, in this way Hadamard proved the prime number theorem (chapter 4), i.e. that the prime counting function is asymptotically equal to the logarithmic integral, and also in this way de la Vallee Poussin derived a bound for the error in this approximation (chapter 5). The Riemann hypothesis would imply a better bound. Chapters 6,7,8,9,11 deal with the pitiful progress towards the Riemann hypothesis, including computational aspects. Chapter 10 tries to hot up the Fourier analysis used in the classical works by putting it in terms of self-adjoint operators and so on. Chapter 12 "Miscellany" includes a proof of the prime number theorem that "is 'elementary' in the technical sense", but, as Edwards admits, it is neither straightforward, nor natural, nor insightful.
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