Customer Reviews

Riemann's Zeta Function, Vol. 58 (Pure and Applied Mathematics)

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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.

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.

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.

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.

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.

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)

While this is a strong mathematical treatment of Riemann's Zeta function, the steps between equations are very terse and not intuitively obvious. A little more time could have been spent filling in steps between equations. This is not a book to read but to study. If you have not had graduate level mathematics, choose another book.

a wonderful exposition full of incredible formulae; a careful account of what Riemann could have thought of when writing his famous paper; the text contains calculations of zeta's non trivial zeroes in an old fashion way but still illuminating. Lacks schema of contours used in calculating
complex integrals but this is a very minor flaw; on the other side, one can find copies of two pages handwritten by B. Riemann which may show how casually this great genius could work and may explain why his followers had such difficulties in proving and following his ideas; the study of zeta means some herculean calculations of which Edwards gives a fine taste; this book is worth the pain to work with; one can compare his first proof of zeta's functional equation with those given in Ahlfors's "Complex analysis" or Tenenbaum's book: "Introduction to analytic an probabilistic theory of numbers"; my favourite formulae are:
1) Von Mangoldt's formula for the psi function with summation over the nontrivial zeroes of zeta.
2) the formula for the product of zeta and gamma as an improper integral and its sequel using hankel contour.
3) the Siegel-Riemann formula for Z(t), this Z function being derived from the values of zeta on the line re(z) = 1/2.
This is the very book to remove oneself on a desert island with (but don't forget a laptop to verify tha accuracy of Backlund's estimations of the fifteen first non trivial zeroes of ... zeta.
By the way, do you happen to know which are the trivial zeroes ?

Undoubtedly the best technical exposition on the Riemann Zeta Function, complete with derivations, proofs, and examples of calculations. As a special treat, it includes a translation of Riemann's original paper on the Zeta function, a photocopy of one page of Riemann's notes, and detailed references to papers of earlier authors of papers and books on the Riemann Zeta function, including a table of values of zeta(½ + it) for t=0 to 50 in increments of 0.2 [Haselgrove, Royal Society Math Tables, Cambridge Univ. Press, 1960].

If someone really want to know more in details about zeta function and its deep implication. Either this or Titchmarsh may do. Both books are excellent in this subject. Titchmarsh's book is more comprehensive but more difficult to read. Edward's one is more approachable and also it include some history and makes it more interersting. Anyway, both books are classics on this field.