Riemann's Zeta Function

Book Description

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.

Customer Reviews:

Riemann Zeta Function & The Riemann Hypothesis.......2007-09-13

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].

A fascinating journey of discovery.......2007-08-18

I had a hard time putting this book down. Who would have thought that for a book about mathematicians and a function? Ok, I am a mathematician and physicist, but still, this really brought to life the exploration and probing of the bizarre and fascinating universe we have created in mathematics.

I highly recommend this book to anyone who likes the history of science and mathematics and is excited by discovery. Even if you only know algebra, I believe that you'll be able to follow the key mathematical concepts and implications.

(Note: four stars is a really, really good rating from me. I haven't given a five star rating yet, which is reserved for literature that is truly mind-blowing.)

Martin's review of "The Zeta Function Book".......2007-08-11

Five stars. Everything you always wanted to know about the zeta function and the Riemann hypothesis but didn't know how to ask. Even though this work was written in 1974, it is still current, as precious little has been achieved toward proving RH. In particular, nobody has come up with a positive epsilon such that all complex zeroes of zeta must lie to the left of Re(s) = 1 - epsilon.

A good guide to Riemann, the prime number theorem, and the Riemann hypothesis.......2006-04-11

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.

Riemann's Zeta Function.......2005-10-24

A vey good synthesis of the state of the art at graduare level.

An Introduction to the Theory of the Riemann Zeta-Function (Cambridge Studies in Advanced Mathematics)

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Analytic Number Theory through Zeta-Function
An excellent resource for those interested in Riemann's

An Introduction to the Theory of the Riemann Zeta-Function (Cambridge Studies in Advanced Mathematics)
S. J. Patterson
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521499054

Book Description

This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.

Customer Reviews:

Analytic Number Theory through Zeta-Function.......2007-01-29

This introductory textbook gives a very good insight of analytic number theory through the special topic of Riemann Zeta-Function. It also includes of course an excellent exposition of its relationship with the Prime Number Theorem and with Riemann and Lindelöf Hypotheses. Moreover, seven appendices provide analytic technical complements needed in the core of the text. Many well-chosen exercises and problems accompany each chapter. I use this textbook in my course on Zeta-Function with great success.

An excellent resource for those interested in Riemann's.......2003-04-12

Zeta function. This book contains a lot of application, theory, and, to my surprise, several practice problems at the end of each section to maximize the learning experience. The chapters are concise and the mathematics is relatively easy follow for those with some experience in special functions.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one.

The Theory of the Riemann Zeta-Function (Oxford Science Publications)

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Complete source of the zeta function
An very comprehesive book for Zeta Function.

The Theory of the Riemann Zeta-Function (Oxford Science Publications)
E. C. Titchmarsh
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Paperback

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ASIN: 0198533691

Book Description

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory,
with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number
theory and classical analysis.

Customer Reviews:

Complete source of the zeta function.......2004-01-19

This is the true encyclopaedia of the zeta function. Although I prefer Ivic, I always have the feeling that Titchmarsh wants to appear brilliant.

This book cannot be criticized because of the amount of time and effort that must have been spent on it. It was update in 1986 by Heath Brown.
It is useless to summarize the contents because it mainly has everything, and most theorems have several proofs and very long comments.
One thing that is missing is more stuff about prime number distributions (for this, check Ingham, Edward's, and a bit of Ivic's).

It never becomes redundant, and it can either be used a source for additional information, as dictionary, or it can be used in a linear way.

An very comprehesive book for Zeta Function........2001-08-22

Titchmarch is well known in the theory of functions, in this book, he described the Riemann's Zeta function in the most comprehensive way. ( e. g. in the topic of functional equation, he quoted 7 methods) I cannot find any other book more comprehensive than this one. ( though in order the theories, you must have some background knowledge and patience ! )

The Riemann Zeta-Function: Theory and Applications

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Riemann Zeta Function - Density of Zeros on the Critical Line
a great update of titchmarsh's book

The Riemann Zeta-Function: Theory and Applications
Aleksandar Ivic
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486428133

Book Description

"A thorough and easily accessible account."--MathSciNet, Mathematical Reviews on the Web, American Mathematical Society. This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estimates, the distribution of primes, the Dirichlet divisor problem and various other divisor problems, and Atkinson's formula for the mean square. End-of-chapter notes supply the history of each chapter's topic and allude to related results not covered by the book. 1985 ed.

Customer Reviews:

Riemann Zeta Function - Density of Zeros on the Critical Line.......2007-09-13

A very good technical exposition on the Riemann Zeta function, and it's applications, with emphasis on zero-free regions, the density of zeros on the critical line, and higher power moments of the Zeta function, but is weak on the computation aspects of calculating the zeros of the Zeta function, and only briefly characterizes the Riemann-Siegel Z formula. It contains copious proofs and technical notes, but also includes three pages of errata that have not yet been incorporated into the text (twenty years later).

a great update of titchmarsh's book.......2003-07-15

this is book is an udpate of the book of titchmarsh (the theory of the riemann zeta function). it covers all the results from 1957 to 1985 (up to atkinson's formula).

if you are looking for stuff about prime numbers, or the prime number distribution, this book will only cover a tiny bit.
there is some stuff devoted to primes, but the main ideas concern only the analytic results of the zeta function, zero free zones and densities, (higher) power moments, exponential integrals, dirichlet problems, approximate functional equation, critical line, riemann hypothesis...

like titchmarsh, ivic normally proves all theorems without assuming riemann or lindelof hypothesis.. there's a chapter where he assumes RH.

Book Description

This ground-breaking work combines the classic (the zeta-function) with the modern (the spectral theory) to create a comprehensive but elementary treatment of spectral resolution. The story starts with a basic but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. The author achieves this by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory. These ideas are then utilized to unveil a new image of the zeta-function, revealing it as the main gem of a necklace composed of all automorphic L-functions. In this book readers will find a detailed account of one of the most fascinating stories in the recent development of number theory. Mathematics specialists and researchers will find this a fascinating work.

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function (Memoirs of the American Mathematical Society)

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Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function (Memoirs of the American Mathematical Society)
Christina Q. He , and Michel L. Lapidus
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings").

Book Description

This volume presents a wide range of results in analytic and probabilistic number theory. The full spectrum of limit theorems in the sense of weak convergence of probability measures for the modules of the Riemann zeta-function and other functions is given by Dirichlet series. Applications to the universality and functional independence of such functions are also given. Furthermore, similar results are presented for Dirichlet L-functions and Dirichlet series with multiplicative coefficients.
Audience: This is a self-contained book, useful for researchers and graduate students working in analytic and probabilistic number theory and can also be used as a textbook for postgraduate courses.

The Riemann Zeta-Function (De Gruyter Expositions in Mathematics)

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The Riemann Zeta-Function (De Gruyter Expositions in Mathematics)
A. A. Karatsuba , and S. M. Voronin
Manufacturer: Walter de Gruyter
ProductGroup: Book
Binding: Library Binding

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ASIN: 3110131706

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The Theory of the Riemann Zeta-Function
E.C. TITCHMARSH
Manufacturer: see notes for publisher info
ProductGroup: Book
Binding: Paperback
ASIN: B000RJJZK0

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The Theory of the Riemann Zeta-Function

Manufacturer: Oxford, 1951
ProductGroup: Book
Binding: Hardcover
ASIN: 0198533217

Book Description

Canes and pipes, especially those made from found materials or by unsophisticated craftsmen, have long been of special interest to carvers and collectors. This guide collects over 200 pen and ink sketches of canes and pipes executed over a period of 35 years by an extraordinary folk artist. These sketches, very impressive in their own right, depict a strange and wonderful group of objects, many with fanciful carvings of mythical or imagined people and animals.

Customer Reviews:

Too weird for me!.......2003-03-24

The carving drawings are ugly and demon-like. If you dig the bizarre, then this is the book for you!

An excellent book of designs and sketches..........1997-02-19

This is an excellent collection of cane and pipe designs! The designs are quite imaginative. While some are impractical, most make very good walking sticks. This is not a book for neophyte cane makers, but a great addition to your shop library

Book Description

This book appears in conjunction with an exhibition organized by The Art Institute of Chicago that focuses on the late work of photographer Edward Weston. Taken between1938 and 1948, these images reveal his shift from his formalist style, characterized by technological virtuosity and innovative compositions, to one that accommodated a greater psychological component. The first photographs of this period date from Weston's return to his spiritual home near Carmel, California, during his second Guggenheim fellowship. He now saw the surrounding coast with different eyes: while he had once focused on details and still lifes, he now found himself drawn to vistas, horizons, the movement of water, and moody atmospheres of elemental power. The seventy-plus photographs in this book, sumptuously printed in tritone reproductions, include--in addition to his images of nature--Weston's powerful portraits of his immediate family, as well as domestic scenes taken in and around his home. Also included is a critical essay exploring Weston's life and work during this period, by David Travis, Curator of Photography at the Art Institute and a longtime specialist in the career of Edward Weston.
Hardcover, 10.5 x 11.5 inches, 144 pages, 100 tritone illustrations.
Exhibition Schedule: Art Institute of Chicago, June-October 2001; San Francisco Museum of Modern Art, March-June 2002.

Customer Reviews:

Rich and dark food for thought.......2002-03-08

This is a catalog for a show currently at San Francisco MOMA, launched in Chicago last year. (Weston came from Illinois and did most of his work in California.) It is essentially a re-edition of Weston's My Camera On Point Lobos, published in 1951 and again in 1968. The major change is text by David Travis replacing excerpts from Weston's daybooks in the original.

The text is intended to humanize someone who is mostly mythical by describing and interpreting events in the last years of his life at Point Lobos. It presents the author's analysis of Weston's career, state of mind and the evolution of his late style. There is little or no new material here and the analysis is strained, but thoughtful.

There are some intelligent comparisons presented of Weston's late and early views of the same subject. As a collection this is not a good introduction to Weston. It is a good final chapter to the Daybooks and a beautiful collection of reproductions. It is also a good companion to Ansel Adams at 100, showing how these two friends viewed many of the same subjects so differently. It would be a good addition to reading Charis Wilson's Through Another Lens, showing many pictures of domestic life including Weston's children, cats, and many of Charis Wilson. There is a lot of "inside baseball" here, both explicit and implied.

There is at least one important image in the show that is not in the catalog and there are many important omissions from the show itself, which make this a poor place to start studying Weston's work. For the record, both Weston and Adams experimented with color in the late 40s, shooting the same images in color and black and white. The color images aren't good but they are a very good way to show why their respective monochrome images are so strong.

It is worth repeating that while the printed images are as good as any you'll see, they are not even close to the 8X10 contact prints in the show. This really matters in Weston's work. If you have a chance to see the San Francisco show, before it is put away for another 10 years, you will also see additional earlier prints from SFMOMA's outstanding permanent collection which put the theme of the show into context that is missing from the book.

This is Weston when he was only satisfying his own search for meaning, not making statements or presenting his vision to the world. These are his final meditations and he knew it. They are by far his richest and most abstract work and worthy of a lot of study.

Edward Weston: The Last Years in Carmel.......2001-12-25

A finely printed book that features more than the regular images that every other book has. The essay is a very worthwhile read. It offers wonderful insites to the photogrpaher at the end of his working career.A real must to any Weston colection of books.

A squirrelly, but talented photographer.......2001-07-20

Edward Weston was one of the most squirelly, yet most talented photographers in the history of the medium - he rarely smiled, wore women's clothes, never learned to drive, married a woman 30 years his junior, lived in a shack in Carmel and loved philandering with Tina Modotti and others. He died with $300 in the bank in 1958, yet his photograph of a Circus Tent went at auction a few years ago for $266,000. His influence on photography and photographers was immense. Two of his four sons, Brett and Cole, became accomplished image makers and his grandson now carries on that same tradition, even living in the same shack on Wildcat Hill in Carmel. This book covers roughly the last 10 years of his photographs 1938-1948. The images are superbly produced and well-chosen but the text was a bit overbearing and heavy on the theory that in the last years Weston was overly concerned with death which was represented in his images. Certainly his images of Point Lobos are a bit dark and morose with pictures of dead trees and pelicans, but that's Point Lobos! During this period he also made whimsical images of his wife wearing a gas mask in the nude and playing a flute while a cat looks on with a surprised glance. Weston was full of LIFE, not death. Thirty years before his death in 1958 he made an image of a corpse at a time when his relationship with his future wife was rosy and he was spending time with his beloved sons. His final work does not seem any more concerned with death than it was in his earlier years. But, forget the text! Photography books are similar to Playboy magazines anyway - we buy them to look at the pictures, not read the text!! This is a terrific book and I can't wait to view the actual images at The Art Institute of Chicago.

the mature artist.......2001-06-25

Not just a great presentation of Weston's last productive years, the essay by the Chicago Art Institute's Curator of Photography provides the best understanding to date of what it means to be a mature artist - and why it was that Weston was viewed by his peers, including Ansel Adams, Minor White, Imogen Cunningham as the consummate photographer, the proof that photography like other forms was capable of synthesizing interior and exterior realities into works of profound emotional and aesthetic power. A great contribution!

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