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Rating: 
Summary: Greatest unsolved math problem, presented unmathematically
Review: "Many people would say that the task I am embarking on by writing this book is doomed to failure." That's the opening line of the book, and I wish one of those people had talked to me before I bought it. Fortunately, I only paid $6 for a used copy. I'd probably take $5, except that I've never resold a math text since my freshman days decades ago.
Sabbagh has basically written a math book for those with neither mathematical knowledge nor the intention to acquire it from his book. It bears the title of the greatest unsolved problem in modern mathematics, but it has the flavor of a survey course for non-technical P.E. or liberal arts majors. He rambles at length about Lives of the Mathematicians, nibbles around the edges of peripheral topics like continued fractions, and spends entire pages on patronizing oversimplifications, like his description of the complex plane.
On at least one occasion, he strays from sloppiness into explicit error: On page 288, discussing infinite series, he observes that the set of prime numbers, like the set of all whole numbers, goes on forever, "yet they can't both have the same 'number' of members (called terms) since one series is included in the other." This is simply, howlingly, false, as it is a *defining characteristic* of an infinite set that it can be placed in one-to-one correspondence with a proper subset of itself. That is, the "number of members" in the set of prime numbers (more rigorously called the "cardinality" of the set) is denoted Aleph-one, the "smallest" infinity; this is also the cardinality of the set of all integers, or the set of all even integers, or the set of all integers consisting only of the digits '7' and '8.'
So, even though it looks like there are only "half as many" even numbers as there are numbers all together, you can simply associate every integer 'n' with the corresponding even integer '2n' and thus every element from one set maps exactly to an element of the other, with none left over. Hence the "number of elements" or cardinality of both sets is equal. Less straightforward mappings work to show equality in the other cases.
Sabbagh goes on for over three hundred pages, and nowhere coherently summarizes the whole Riemann Hypothesis and its mathematical underpinnings, which he could certainly do in maybe one chapter, and which I could probably do here ultra-briefly if I had a charge number for a couple of hours.
I love math--and loved it recreationally back in junior high, long before I had any degrees or great knowledge of it. This is the first and only book written for adults I've found this useless. Think about a documentary on some obscure detail about the sex habits of a tribe you've never heard of on the other side of the planet, edited for a PG rating and broadcast in your second language. That's how much of the Riemann Hypothesis you get here.
Rating: 
Summary: Seekers of the truth
Review: 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.
Rating: 
Summary: WOW -- SOME PEOPLE REALLY HATED THIS ONE
Review: I have not yet read PRIME OBSESSION which at least one reviewer recommends instead of this book, but I at the very least enjoyed this one. Unless your math background includes calculus and at least an introductory level course in complex analysis, you are not really going to understand what the Riemann Hypothesis says in any deep way. That's just the way it is. (Similar problems exist in any book written about quantum theory, say.) However, I thought the author did a good job of giving a non-mathematical reader a feel for what the hypothesis is with his "addresses in New York" metaphor. The fascination of this book is its introduction to a wide variety of mathematicians who are working on proving the hypothesis and a realistic idea of a) how much time it takes to "do math" and b) the one-mindedness that a problem like this creates in the people who study it. This is a relatively low stress popular math book intended for the general reader who might otherwise never pick up a book about a math problem. I thought the author succeeded in reaching that audience quite well.
Rating:
Summary: Forget it !!
Review: 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.
Rating: 
Summary: good but not good enough
Review: 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.
Rating:
Summary: Let us not discredit this fine attempt
Review: It would be very tempting for the mathematically-inclined to criticize this book for what is, to the adroit reader, its laborious treatments of some very basic mathematics. Many already have. But do not overlook this book's value, especially if you are not one of the mathematical elite. What Sabbagh has done is to take a rather unapproachable topic and put it in the terminology of the layman. Unlike many other texts which fail in this endeavor (by lapsing into terminolgy which has not been sufficiently defined or concepts which are not solidly built up within the volume), Sabbagh is thorough to the point of pedantically assuring that the reader can follow every step of the way. Some of this pedantry would be better directed to ensuring absolute accuracy, true, but the bulk of this book guides it toward achieving its author's intent.
The sometimes amusing anecdotes are worth the price of the book, and let's face it - any book purporting to be accessible to the man on the street that contains an explanation of eigenvalues has got some guts behind it, and, in my view, "The Riemann Hypothesis" has a great deal of merit as well.
Rating:
Summary: Leaves the reader somewhat disappointed.
Review: 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!
Rating:
Summary: good but not good enough
Review: This book is really about the author's view of how mathematicians think and live in the current day. There are oodles of little tidbits of "weird mathematicians". You get some idea of what the RH is about, but it's not _really_ the point of the book. Unfortunately, the focus on De Branges overcorrects the math establishment's treatment of him, which, though unfortunate, is partly justified (he announced yet another proof today or yesterday - which may be correct this time). Not recommended overall, but definitely gives a slice of life from the sciences that acknowledges both human failings and sublime thoughts.
Rating:
Summary: The Riemann Hypothesis ... sorta revealed
Review: Well, anyone willing to try to explain the RH to the "person in the street" deserves some credit, regardless of the success of the effort.
In my view, Sabbagh's book will not give one much of a feel for the mathematical machinery involved with the RH, even at a very elementary level. There are some infelicities here and there and, at times, Sabbagh's own understanding is in doubt - even at his "non-mathematician" level. This is nothing to be ashamed of ... there are many professional mathematicians that are afraid of number theory and consequently have little or no understanding of the RH either.
I'd recommend Derbyshire's Prime Obsession as a much more informative and better read for the novice. The next step up is a giant one and few are prepared to make it. For those interested, a reasonable and gentle book to consider is "The Prime Number Theorem" by G. Jameson. The latter is an excellent introduction to analytic number theory including the RH. Ok, if you want to go all the way, try Edwards' "Riemann's Zeta Function" ... but before you do, fasten your seatbelt!!
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