Riemann's Zeta Function [Paperback]

Harold M. Edwards (Author)

Book Description

ISBN-10: 0486417409 | ISBN-13: 978-0486417400 | Publication Date: June 13, 2001 | Edition: Dover Ed

Superb study of one of the most influential classics in mathematics examines the landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics.

Product Details

• Paperback: 330 pages
• Publisher: Dover Publications; Dover Ed edition (June 13, 2001)
• Language: English
• ISBN-10: 0486417409
• ISBN-13: 978-0486417400
• Product Dimensions: 8.5 x 5.4 x 0.7 inches
• Shipping Weight: 12 ounces (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (16 customer reviews)
• Amazon Best Sellers Rank: #175,067 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

85 of 86 people found the following review helpful:

5.0 out of 5 stars Good complement to Ivic and Titchmarsh, January 18, 2004

By 

J. N. M. ROBLES "Nicolas M. Robles" (London, UK) - See all my reviews
(REAL NAME)   

This review is from: Riemann's Zeta Function (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:

5.0 out of 5 stars This book is great, July 13, 2005

By 

MathGeek741 "jackgoodman2" (Maryland, USA) - See all my reviews

This review is from: Riemann's Zeta Function (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.

45 of 47 people found the following review helpful:

5.0 out of 5 stars Excellent for experts and the casual mathematician alike, September 19, 2005

By 

Bachelier ""1004"" (Ile de France) - See all my reviews

Amazon Verified Purchase

This review is from: Riemann's Zeta Function (Paperback)

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.