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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, by Karl Sabbagh
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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.
- Sales Rank: #1626400 in Books
- Published on: 2003-04-30
- Original language: English
- Number of items: 1
- Dimensions: 1.15" h x 5.82" w x 8.48" l,
- Binding: Hardcover
- 304 pages
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
About the Author
Karl Sabbagh is the author of six books, most recently A Rum Affair (FSG, 2000). He lives near Stratford-upon-Avon in England.
Most helpful customer reviews
18 of 19 people found the following review helpful.
Ready for prime time
By Gary C. Marfin
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.
16 of 37 people found the following review helpful.
Approachable, but Lost my Trust
By Mike Blaszczak
As a math hobbyist, I've been very eager to learn more about the Riemann Hypothesis. I ordered this book a couple weks ago, and fell ill this week -- so I had some time to spend reading.
I didn't get very far before I found a rather glaring error. On page eighteen, the author explains one method for showing that there's no largest prime number. He asserts that:
(2 * 3 * 5 * 7 * 11 * 13 * 17 + 1) / 7
is equal to
2 * 3 * 5 * 7 * 11 * 13 * 17 + (1/7)
which is not correct. The left-hand term actually evaluates to
2 * 3 * 5 * 11 * 13 * 17 + (1/7).
Maybe I'll revise my review after I read more; perhaps the book really does deserve my trust. But after an early blunder like this in a simple topic, maybe I won't read any further.
Less astute readers might not have caught this and later become confused. Maybe, in the deeper topics, I won't be able to proof the text and become confused or disillusioned by other errors.
7 of 8 people found the following review helpful.
Seekers of the truth
By Donald B. Siano
I have an undergraduate degree in mathematics from a long time ago, but haven't done a whole lot with it. Nevertheless, I gained something of an appreciation for the subject and am always interested when something important enough happens that it gets into the popular press. So naturally enough, I have been aware of the number one unsolved problem of mathematics, the Riemann hypothesis, and have followed the sporadic claims of its resolution over the last few decades.
Mr. Sabbagh's popular treatment of the problem in this book was a delight for me to read. He explains the hypothesis very clearly in a way that really doesn't even require any specialized knowledge of any arcane area of mathematics. Though great, this is not the primary virtue of the book. Rather it is his effort to reach out to the dozen or so mathematicians who are actively working on the problem who might have a hope of finally, after about a century and a half, of proving it. The reader is thus led to some appreciation of the world of the professional mathematician, with all of its human hopes and jealousies, striving to achieve a legacy that will outlive themselves. Sabbagh interviews them, some of them several times, attends their seminars, and listens for the inside dope that might show that someone somewhere is onto something.
The book is engaging, and I found it impossible to put down. It has lots of anecdotes, asides, and curiosities along the way to liven up the story. It is brutally honest in its portrayals of the principle characters. The writing style is lively, and the math is easy to follow. And it tells a story of man at his best--striving for progress, precision and truth. Quite the opposite of so many charlatans of the academy today, who seem to revel in ambiguity, imprecision, and political correctness.
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