Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics | John Derbyshire | detailed and accessible introduction to a difficult unsolved problem in mathematics

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Prime Obsession: B...

	Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John Derbyshire Plume, 2004 - 448 pages average customer review:based on 83 reviews view larger image
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highly recommended

First-rate!!

Somewhere towards the beginning of PO, JD remarks that the reader who doesn't understand (the general outlines of) the Riemann Hypothesis (RH) after reading PO will never understand it. I am inclined to agree. The RH is very tough going -- clearly worlds tougher in its rigorous mathematical treatment than in the elementarized version that JD presents. Even in the latter version it's pretty tough going, though, especially in the closing chapters of PO, which increasingly -- and necessarily -- fall back on 'you'll have to trust me, there really is a nifty (albeit exceedingly complex) way of tranforming this function into that one'-type formulations. There is no implied criticism here. JD does a superb job of steadily piling concept on concept, formula on formula, and providing the attentive reader with deep insights into the actual mathematics of the RH. JD is a wonderful guide. He writes with wit and enthusiasm, and a tremendous sensitivity for the expository (i.e., logical) requirements of an interested but mathematically in- or under-experienced readership. As I reached the end of PO, certain arguments eluded me still, but I did feel that I essentially 'got' the RH. Moreover, I emerged with a deep appreciation for the awe, fascination, and delight that mathematicians who have pondered or worked on this problem feel towards it. One day I hope to tackle some of the major treatments of the RH. Towards that end, I have no doubt that I will return to PO for insights and explanations again and again. PO is brilliant! Bravo, JD!!

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detailed and accessible introduction to a difficult unsolved problem in mathematics

Prime Obsession is about the Reimann Hypothesis. However, just stating the Reimann Hypothesis in a comprehensible way is difficult. There is a lot of background material you need to know to even understand the problem statement. The author succeeds at presenting this material admirably. Odd chapters have math and even chapters have the history of the problem. I greatly enjoyed this book. Although the going is tough sometimes, the author's prose is exciting and highly readable.

The best book on the Riemann hypothesis

The Riemann hypothesis is now the most famous unsolved problem in mathematics. First stated by Bernard Riemann in 1859, it has resisted all attempts at solution. Most mathematicians consider it to be true and all computer computations have affirmed this. Simply stated it is:

The real part of any non-trivial zero of the Riemann zeta function is ½.

The Clay Mathematics Institute has offered a prize of one million dollars to the first person who is able to resolve the issue.
In this book, Derbyshire gives the complete history of the problem, explaining it in a manner that should be accessible to all. It reads like a mystery novel with no unmasking of the culprit. A complete background of the problem and all of the clues found to date are listed, yet the best that can be said is that all evidence points to it being true. Although there have been some statements skeptical of the truth of the hypothesis, they have been rare and often qualified.
This is without question the best book on the Riemann hypothesis written so far. It would make an excellent supplemental text for courses in the history of mathematics and can be read just for fun by professional mathematicians.

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excellent history and overview, with some details omitted

The Riemann hypothesis is one of the greatest unsolved problems in mathematics, with a $1,000,000 bounty for its solution. Although the Riemann hypothesis is intimately related with prime numbers, one of the most elementary notions in mathematics, the statement of the Riemann hypothesis is much more difficult to understand. This book attempts to explain the meaning, significance, and history of the Riemann hypothesis, while assuming minimal math background, as one might get from a good high school education. In the introduction the author provocatively writes, "if you don't understand the [Riemann] Hypothesis after finishing my book, you can be pretty sure you will never understand it". I think that what the author means here is that he gives the most elementary possible treatment of the subject. However I think that his treatment is a bit TOO elementary for one to REALLY understand what the Riemann hypothesis says. For example, although the Riemann hypothesis concerns the zeta function of complex numbers, I don't think he ever actually defines the zeta function of a general complex number. Nonetheless, given the level at which he chooses to write, I think he does about as well as one could hope to convey the rough idea of what is going on from various points of view.

I should say here that my perspective is that of a professional mathematician who does not know much about the Riemann hypothesis.

The format of the book is interesting. Roughly speaking, odd numbered chapters explain math, while even numbered chapters present history, although the history chapters have a bit of math in them and vice-versa. I found the history chapters quite enjoyable. The math chapters are a mixed bag. Some parts review basic math; I skipped over those. Other parts introduce the key players in the Riemann hypothesis. There is excellent use of pictures and numerical evidence to motivate various theorems and conjectures. Because every other chapter is devoted to history, and because so much math needs to be introduced, it takes a long time to get to the heart of the matter, which appears near the end of the book. There is also a major speedup near the end, with more and more mathematical details omitted.

If the proof or disproof of the Riemann hypothesis is a mountain summit, and if the statement of the Riemann hypothesis and its connection with prime numbers is base camp, then this book leads us on a walk through the foothills while discussing the history of various people who have explored this mountain, and then makes a mad dash for base camp which ends up requiring a helicopter rescue.

The book overreaches a bit in discussing speculative connections of the Riemann hypothesis with physics. In doing so the book spends some time discussing eigenvalues of random matrices, without giving any clue what an eigenvalue is, so I don't know how much the target audience is going to get out of this. I think it would have been better to just say, "here is the distribution of the first 500 zeroes of the zeta function, here are 500 random numbers, notice that the spacing looks different, a similar spacing appears in physics via eigenvalues of random matrices, no one knows if this is a coincidence or if there is an actual connection, here is a reference where you can read more about this."

It would have been nice to include Riemann's original paper on the subject as an appendix. While only experts can understand this, it is pretty inspiring. (A translation with a few notes can be found for example in "God created the integers".)

Despite my nitpicking above, I heartily approve of this book, because I think that mathematics needs more popularization, and this book has done an admirable job with difficult material.

Also, silly as this is to say, I have become much more of a Riemann fan after reading this book. I use Riemann integrals, Riemann surfaces, Cauchy-Riemann equations, Riemannian geometry, etc. on an everyday basis, but I never paid attention to the fact that they all have the same name in them. His accomplishments are brilliant and spectacular and completely transformed mathematics.

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