The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics [Hardcover]

Karl Sabbagh (Author)

Book Description

Publication Date: April 30, 2003

An engaging, informative, and wryly humorous exploration of one of the great conundrums of all time

In 1859 Bernhard Riemann, a shy German mathematician, wrote an eight-page article giving an answer to a problem that had long puzzled mathematicians. But he didn’t provide a proof. In fact, he said he couldn’t prove it but he thought that his answer was “very probably” true. From the publication of that paper to the present day, the world’s mathematicians have been fascinated, infuriated, and obsessed with proving the Riemann Hypothesis, and so great is the interest in its solution that in 2001 an American foundation put up prize money of $1 million for the first person to demonstrate that the hypothesis is correct.

The hypothesis refers to prime numbers, which are in some sense the atoms from which all other numbers are constructed, and seeks to explain where every single prime to infinity will occur. Riemann’s idea—if true—would illuminate how these numbers are distributed, and if false will throw pure mathematics into confusion.

Karl Sabbagh meets some of the world’s mathematicians who spend their lives thinking about the Riemann Hypothesis, focusing attention in particular on “Riemann’s zeros,” a series of points that are believed to lie in a straight line, though no one can prove it. Accessible and vivid, The Riemann Hypothesis is a brilliant explanation of numbers and a profound meditation on the ultimate meaning of mathematics.

Editorial Reviews

From Publishers Weekly

With Fermat's Last Theorem proved, the Riemann Hypothesis has become math's most glamorous unsolved problem, and has spawned a growing literature seeking to explain it to lay readers. Unfortunately, this curious genre is overshadowed by the fact that the hypothesis itself is incomprehensible to anyone without a Ph.D. Sabbagh, author of A Rum Affair, struggles manfully with this problem, and gives impressively lucid explanations of such preliminary subjects as prime numbers, logarithms, infinite series, algebraic equations and matrices. But even with all this background, the hypothesis remains such an opaque abstraction that, at one typically baffling juncture, the author throws up his hands and instructs readers to either "sign up for a few months of complex analysis and number theory, and then pick up the book again in a year or two" or else just "take it on trust." To help elucidate the material, Sabbagh includes many lengthy excerpts from interviews with mathematicians, who, he claims, "see truths with a clarity that is sometimes breathtaking," but these rambling, obscure commentaries ("what's going to probably happen for the real Riemann Hypothesis is there's going to be another blob and there's going to be a function that turns the blob into itself") are not necessarily very helpful. Sabbagh can be a gifted expositor of mathematics when he sticks to more tractable topics, but when it comes to the Riemann Hypothesis, he offers readers veneration instead of understanding. B&w illustrations and graphs.
Copyright 2003 Reed Business Information, Inc.

From Booklist

Bernhard Riemann would make any list of the greatest mathematicians ever. In 1859, he proposed a formula to count prime numbers that has defied all attempts to prove it true.This new book tackles the Riemann hypothesis. Sabbagh introduces contemporary mathematicians who are working on the problem, one of whom claims, to professional skepticism, to be on the verge of vindicating the hypothesis. Another is working away in search of a single counterexample that would refute it. Such pursuits, which often consume mathematicians' entire lives, may seem incomprehensible or even pointless to the innumerate--but that's a prejudice brilliantly dispelled through Sabbagh's interviews, which are interwoven with his not overly numerical tour of the hypothesis. The drive and competitiveness of mathematicians clearly emerge from Sabbagh's narrative.

Gilbert Taylor
Copyright © American Library Association. All rights reserved

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Product Details

• Hardcover: 304 pages
• Publisher: Farrar, Straus and Giroux; 1st edition (April 30, 2003)
• Language: English
• ISBN-10: 0374250073
• ISBN-13: 978-0374250072
• Product Dimensions: 8.4 x 5.8 x 1.2 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 3.2 out of 5 stars  See all reviews (21 customer reviews)
• Amazon Best Sellers Rank: #1,112,521 in Books (See Top 100 in Books)

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17 of 18 people found the following review helpful:

5.0 out of 5 stars Ready for prime time, November 30, 2003

By 

Gary C. Marfin (Sugar Land, Texas USA) - See all my reviews
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This review is from: The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Hardcover)

I think potential readers of this intriguing book need to bear in mind the following: (1) you do not need to understand Riemann's hypothesis to enjoy this book and (2) Mr. Sabbagh does a very fine job of outlining Riemann's hypothesis in layman's terms. Riemann's hypothesis is not easily grasped; what Sabbagh wants to do is to enhance your understanding of it. There is no pretense here that the hypothesis in all its complexity is being conveyed. In fact, near the book's end, he concedes that "you know almost nothing [about R's hypothesis] compared to what there is to know. The hypothesis itself is an outcome of Riemann's zeta function which is the sum of the series 1 + 1/2^s +1/3^s...1/n^s, which means 1 + 1/2^a+ib + 1/3^a+ib (where i is an imaginary number). All sorts of values are possible, but the values of interest center on the Riemann zeta function when it becomes zero. These zeroes, as its turns out, fall on what is known as the "critical strip" and their graph is linked to the fluctation of the primes, which are themselves the building blocks for all the other numbers. The hypothesis is that all the "significant" zeroes line on the critical strip. The proof has become the Mount Everest of mathematics, but it remains unscaled. Many mathematicians, who perhaps found the hypothesis disarmingly approachable, have died before reaching the summit.

Sabbagh does want you to understand the hypothesis, but he is also trying to delve into the community of mathematicians generally -- what they are like as people -- in an effort to make them more accessible as well. Inevitably, in this area, Sabbagh often reads like an anthropologist documenting the ritualistic "abnormalities" of some primitive Amazonian sub-culture. What I found surprising is not that Sabbagh finds that the thought processes of mathematicians rarely intersect with that of non-mathematicians; rather what I found striking were the similarities with the "rest of us." They can be collaborative yet guarded, brave yet insecure, intuitive but distrustful of intuition. Several he finds are lousy at simple computations (but brilliant on abstractions). They are a colorful lot, but they are not high IQ aliens from another world. The portrait of Louis de Branges is especially fascinating and forms a strong sub-plot within Sabbagh's text.

I don't plow through many books like this, but I do recommend The Riemann Hypothesis. Like Sarah Flannery's "In Code" (which has an excellent chapter on prime numbers), The Riemann Hypothesis is suited for, and ought to be attractive to a wide audience.

9 of 9 people found the following review helpful:

4.0 out of 5 stars Pleasant enough, February 16, 2004

By 

wiredweird "wiredweird" (Earth, or somewhere nearby) - See all my reviews
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This review is from: The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Hardcover)

I want to call this a "biography", but the Riemann Hypothesis isn't biological. It's almost take on a life of its own, though - maybe the term really does apply.

In any case, this is a very enjoyable book about the history of the hypothesis. In many ways, this book is more about the people who pursue that elusive proof. That small, distinguished crowd includes the reticent and the outspoken, the loners and the social thinkers, the meticulous and those who think by leaps and bounds. Sabbagh has a strong emphasis on the living mathematicians who hunt this elusive quarry. He has spent long hours interviewing these mathematicians and watching them at their work. At bottom, this may be a book about intellectual passion and the people for whom its reward is real.

The book contains a few disconcerting mis-statements:
-- one says that plutonium occurs naturally - on Earth, it does not,
-- another on p.11 makes a statement about prime factors of the number 60 (I'd believe that same statement about all of 60's factors, including non-primes), and
-- a third on p.143 appears to have applied parentheses incorrectly in describing Skewe's number.
None of these, by itself, affects the main thrust of the book. Still, they leave me wondering about every fact I read. When I find such errors, I have to wonder how many I didn't find, ones that I don't have the information to check.

Because of the book's emphasis on the people dedicated to the hypothesis, there is no one place where the hypothesis' history is laid out in full and in order. That's small enough loss, if you accept that the book's topic is really mathematicians and not mathematics. The author does give a brief and clear statement of the problem itself - that takes math at the level of high school calculus to understand, but the reader won't be punished for skipping past its details.

This book has real nerd appeal (I like it). It's a readable case study of a famous problem and of the people tracking it down. It won't really expand anyone's intellectual horizons, but there are lots worse ways to spend a few hours. Despite flaws, I found this book quite enjoyable.

9 of 9 people found the following review helpful:

3.0 out of 5 stars Leaves the reader somewhat disappointed., January 4, 2004

By 

"jayjina" - See all my reviews

This review is from: The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Hardcover)

Leaves the reader somewhat disappointed.

I picked up this book with great expectations, having read the publisher's publicity. To be frank, I was left disappointed. The book tells the reader very little about the wonderful and mysterious character of the Riemann hypothesis and leaves both mathematical novices and those who know about the intricacies of higher Mathematics dissatisfied. This is indeed a pity!

Having said this, Mr Sabbagh's story is eminently readable and enlightening. The book has many sections that are in effect a diary of the conversations with various Mathematicians. These give an insight into the thought processes, passions, motivations, and rivalries that exist in the select community of Number Theorists. The pen portraits of the main protagonists is quite interesting even though it sheds little light on the character of the Riemann hypothesis and how it enthrals those working on its proof.

The toolkits covering a set of brief synopsis of Infinite series and the Euler identity should be useful to the lay (but Mathematically capable) reader, but the appendix on the De Brandes proof is rather obscure.

Overall, an OK book if the reader wants a gentle introduction to the subject and act clever in passing conversation at parties, but, sadly, this book fails to educate and enlighten in the real sense!