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

Oct 15, 2009 Jason rated it 3 of 5 stars · review of another edition

You remember the smartest kid in your high school calculus class? Remember the math major in your college dorm, the one doing advanced physics with more Greek symbols than Roman numerals? Both brainiacs at the time, right? Well, the book Prime Obsession deals with mathematical concepts magnitudes of order more complex than those brainiacs could ever wish to comprehend. John Derbyshire describes the Riemann Hypothesis (RH) and the mathematical titans that have tried unsuccessfully to prove the hy...more You remember the smartest kid in your high school calculus class? Remember the math major in your college dorm, the one doing advanced physics with more Greek symbols than Roman numerals? Both brainiacs at the time, right? Well, the book Prime Obsession deals with mathematical concepts magnitudes of order more complex than those brainiacs could ever wish to comprehend. John Derbyshire describes the Riemann Hypothesis (RH) and the mathematical titans that have tried unsuccessfully to prove the hypothesis since 1859. Open the book and you'll be transported to the rarefied air of the Prime Number Theorem and the notorious zeta function.

Derbyshire does a nice shuffle between chapters, alternating between math concepts and biographical background. This does two things: it guides you slowly through the math of Greek symbols and presents the human side of the computational oracle known as Georg Friedrich Bernhard Riemann. I'm not a math guy. I don't even enjoy math. However, after struggling in several higher level calculus classes in college, I have a humble appreciation for and stark awe of folks who can turn, transform, and invert insanely dense and convoluted proofs and theorems. I also realize math is hidden behind many of the technological advances we take for granted as our lives get easier. I don't stop to make the connections between prime number theory and secure internet transactions, but I know it's there, below, hidden, operating, critical to my life.

When I saw several books about the RH, I read a dustcover and, at once, needed to be educated about a theorem that's remained relatively unchanged, and is still only as advanced as when it was presented by Riemann 150 years ago, despite Cray-1 supercomputers crunching possible solutions for 40 years and programmed by myriad math geniuses produced by the 20th century. If it hasn't been solved, then I thought—having never heard of the RH before—Riemann was either a genius among geniuses, or he stumbled upon something mankind was perhaps not to discover for many, many generations to come. Either way, I figured the book would give me a few buzz words to finesse around a cocktail party in the extraordinary event math was discussed over finger foods and wine.

I picked up some buzz words, indeed, and I can use them to parry myself through the initial awkward questions about math. However, I'd have to duck and run from the party because Derbyshire didn't do the best job making the RH understandable. The author is a mathematician, and though he promised to use nothing more advanced than calculus to explain the RH, he was quickly over my head. I know general calculus (quadratic polynomials, convergent & divergent limit functions, matricies, harmonic series, derivation & integration). But, let me walk you through what Derbyshire introduced to me in some 350-odd pages.

--pi(N), but not the pi that is 3.14159....
--the Euler-Mascheroni number
--the Rieman Hypothesis: All non-trivial zeros of the zeta funtion have real part one-half
--the Golden Key
--the sieve of Eratosthenes
--the Prime Number Theorem, pi(N)~Li(N)
--complex conjugates
--zeta function critical strip
--Gram's zeros
--Riemann surface
--value plane from the critical line
--Big Oh and Mobius Mu
--Matricies (eigenvalues, trace, characteristic polynomials of)
--operators
--Guassian-random Hermitan matricies
--Guassian Unitary Ensemble
--Chaos theory
--And many, many, many complex, irrational formulas without Roman numerals

To give Derbyshire credit, what Prime Obsession attempts to do is chronologize the different attempts by which mathematicians have tried to solve the RH. Our 20th century boys have not been completely stumped. They've made minor inroads, cracking stone off the monolithic RH, using tactics, techniques, and procedures that are au courant at the time. The book shows calculus, geometry, analytical theory, physics, quantam theory—all specific modus operendi that offer tools of analysis, and which could each stand to benefit from a solution to the RH.

Still, this book is not for the math novice. Also, I think Derbyshire fell short presenting, in plain language, what exactly the RH would prove, and if solved, how it would change the world of mathematics. 3 stars for a solid book, but it won't appear on anybody's favorite list.

New word: ruction

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Feb 19, 2013 Manny rated it 4 of 5 stars

Shelves: science

The best popular mathematics book I can recall reading. I had heard about the Riemann Conjecture a zillion times, and never understood what the fuss was about. After going through this book, it all made sense! Requires college-level math, but if you have that, can't recommend too highly.
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An anecdote from Lambert's biography of Georges Lemaître which may amuse mathematicians. At one stage, young Lemaître was being supervised by the famous number theorist de Vallée-Po...more

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Jul 09, 2007 Elio rated it 4 of 5 stars

Although I find this author's political views repellent, I really enjoyed this book. He takes an extremely esoteric mathematical puzzle and shows how it emerges organically starting from the simple math we learned in high school. He also provides several excellent character sketches of famous mathematicians who made the key discoveries that allowed the Riemann Hypothesis to come into being in the first place. Most importantly, Derbyshire manages to convey the sense that the field mathematics is...more

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Jan 29, 2013 Karl rated it 4 of 5 stars

Shelves: mathematics, scients

Excellent pop-science work. As a mathematician, it was a little simplistic for me (although as a mathematician who did not specialise in number theory I still got a lot out of this book; mathematics is an evolving and complex discipline with many subfields for specialisation and, as Derbyshire points out in this book, it is no longer possible for a mathematician to be an expert in all fields of mathematics anymore. There's simply not enough time in the day); equally, I can see that a non-practit...more

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Jul 16, 2011 Bryan Higgs rated it 4 of 5 stars

Shelves: mathematics, history-mathematics

This book is one of several books on a mathematical topic, ostensibly for laypersons. The topic in this case is the Riemann Hypothesis, which is one of the -- perhaps THE -- most important unsolved problems in Mathematics. The style and layout of the book follows one that I have seen in other such books, where the chapters alternate between the history and personalities and social and political context for those involved in trying to solve the problem, and an explanation of the mathematical topi...more This book is one of several books on a mathematical topic, ostensibly for laypersons. The topic in this case is the Riemann Hypothesis, which is one of the -- perhaps THE -- most important unsolved problems in Mathematics. The style and layout of the book follows one that I have seen in other such books, where the chapters alternate between the history and personalities and social and political context for those involved in trying to solve the problem, and an explanation of the mathematical topic or related topics.

In the case of the Riemann Hypothesis (RH), the mathematical explanations are not easy. The hypothesis itself is not easy to understand. The author defines the RH on page xi in the book's prologue: "The Riemann Hypothesis: All non-trivial zeros of the zeta function have real part one-half." He then spends a considerable part of the book explaining what that means. This involves introducing number systems, complex numbers, some integral calculus, and a lot of manipulation of infinite series.

I have a solid background in mathematics (my Ph.D. is in Elementary Particle Physics), albeit I'm on the rusty side, not having used my math much in recent years. I found some of the math explanations somewhat challenging, so I'm not sure what a genuine "intelligent layperson" would make of them. I found them interesting, and sometimes innovative, but towards the end I was growing tired and losing interest.

The chapters on the mathematicians, their history and social and political context were more interesting (and I am one who found history in high school to be boring in the extreme!)

On the whole, I liked the book, but I have also read Marcus du Sautoy's "The Music of the Primes", which is also focused on the RH, and I prefer that book.

Towards the end of this book, one starts to wonder what makes the RH so important to mathematicians. In Chapter 22, the author addresses the question "What use is it?", and then goes on to say that he, along with most pure mathematicians, don't really care whether there are applications or could be. But that is not the same as the question "Why is it important in mathematics?", and I felt after finishing the book that this latter question hadn't really been answered (unless you consider the entire contents of the book an answer; if so, it certainly isn't a succinct answer!)(less)

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Oct 03, 2010 Josh Hamacher rated it 3 of 5 stars

Reading this reminded me of how much I always enjoyed math back in college. The chapters alternate; odd chapters are a mathematical-oriented exposition of what the Reimann Hypothesis is, what it means, and what it implies. Even chapters give an historical overview of the key players, places, and events in the search.

I can't say I kept up with all of the math. The first third was a review, the second third I felt like I should understand because I had it in college but hadn't used since, and the...more

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Nov 27, 2012 Theresa Leone Davidson rated it 2 of 5 stars

If you know of the Riemann Hypothesis and how no one has been able to solve it, or if you have a whole lot of interest in mathematics, particularly calculus, or if you are the type of person who enjoys solving difficult math problems just for fun, then you would probably enjoy this book. I did not learn to love math until college, which I have written about in past reviews, but I do not have an inordinate amount of love for the subject, and this book I found to be tedious and overwhelming. I do...more

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Feb 14, 2011 Jbunn rated it 5 of 5 stars

This is an outstanding overview of the Riemann Hypothesis and its relation to the Prime Number Theorem, from both a historical and a mathematical perspective (in alternating chapters). If you have a basic knowledge of calculus, including sequences and series, along with just a smidgen of complex analysis, then you should have no difficulty following all of the math in this book.

Derbyshire's mathematical exposition is quite good, especially for an author who is not a practicing mathematician and...more

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Feb 18, 2013 Emma Glaisher rated it 5 of 5 stars

This was pitched absolutely right for me. OK, by the end I lost the plot (fields, operators... ) but he handheld beautifully and talked the reader through some actual calculations, and totally explained how you get from the zeta function to the Euler product (a result which Marcus du Sautoy simply states with no explanation, leaving me totally baffled).

Reminded me of matrices and eigenvalues, which we did for maths O level back in the 70s! Need to go back and look more at this.

Perfect complement...more

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Dec 12, 2011 Flora rated it 2 of 5 stars · review of another edition

Shelves: nonfiction, books-that-take-me-to-other-countri

The author alternates between more narrative-type chapters and mathematical explanation chapters, and explains early on that if you don't get the math you can just read every other chapter. I tried to follow, but somewhere in the last six chapters I just fell and couldn't keep up.

Fabulous writing. Awesome footnotes. Thanks for jogging my memory about some long-unused keys on my scientific calculator, and for the lovely mathematicians' biographies. But at the end of the day, yeah, it was kind of...more

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Jun 11, 2010 Andy rated it 4 of 5 stars

Shelves: math, nonfiction

There are famous math problems that are easy to explain but difficult to solve, such as the four-color map problem or Goldbach's Conjecture; the Riemann Hypothesis is unfortunately not such a problem. Prime Obsession is author John Derbyshire's attempt to explain the RH in simple terms and to illustrate its place and importance in the history of mathematics. It's not an easy task, and I think what Derbyshire has written is suited for a relatively narrow audience of people: those who took some an...more There are famous math problems that are easy to explain but difficult to solve, such as the four-color map problem or Goldbach's Conjecture; the Riemann Hypothesis is unfortunately not such a problem. Prime Obsession is author John Derbyshire's attempt to explain the RH in simple terms and to illustrate its place and importance in the history of mathematics. It's not an easy task, and I think what Derbyshire has written is suited for a relatively narrow audience of people: those who took some analysis or at least calculus in school but who didn't go on to enough pure maths to have already learned about the RH.

I fall under this category. The book did succeed in explaining the RH to me. But I do not think it imparted an adequate appreciation of why it is such an important problem. Some of the other results in the book, such as Euler's theorem (the "Golden Key") or the existence of Littlewood inversions, seemed to me to be more interesting or elegant. In particular, I do not know except in brief passing reference what mathematics would depend on the RH, and I do not know what the ramifications of a disproof would be. Towards the end Derbyshire goes very quickly through newer, complicated physics-based advances regarding the RH, leaving little possibility for deep understanding; I would have gotten rid of this section in favor of deeper coverage of the RH's consequences, even if this meant that we wouldn't be covering math newer than the 1930s.

Nevertheless I generally enjoyed the book. But then again I fall in the narrow audience mentioned above. If you are more mathematically advanced than I am, you may find this book entertaining but too basic. If you are less so, you may not even understand the RH properly at all, which is supposed to be the main purpose of the book. Rather than communicating the intuition and big picture, Derbyshire sometimes takes more of a procedural, step-by-step approach which he "babies" a bit to help along laypeople. This does help get the casual reader from point A to point B but is not going to leave a framework of true understanding in the reader's mind.

The odd chapters are more strictly mathematical and the even chapters are more historical. The history chapters for the most part can be digested and enjoyed by anyone, so even for relative math-phobes, half of this book will be pretty readable. Derbyshire does a very good job here of depicting the real people behind the theorems, not only with regard to their personalities but how their lives fit into the greater context of world history. I found the discussion of the great mathematicians' patrons to be particularly good. (less)

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Apr 10, 2011 Fraser Kinnear rated it 4 of 5 stars

Shelves: math, history

Prime numbers are powerful things. If you multiply one or more primes together, you can create any other positive integer that's bigger than one. And we suspect that every even positive integer greater than 2 is the sum of two primes.

But primes are strange as well - there doesn't appear to be any order to their appearance. The higher you count, the less often you run into them and you'll never stop seeing them. But can we tell when the next one will occur? In other words, is there some sort of p...more Prime numbers are powerful things. If you multiply one or more primes together, you can create any other positive integer that's bigger than one. And we suspect that every even positive integer greater than 2 is the sum of two primes.

But primes are strange as well - there doesn't appear to be any order to their appearance. The higher you count, the less often you run into them and you'll never stop seeing them. But can we tell when the next one will occur? In other words, is there some sort of pattern to their appearance? And could that pattern tell us something fundamental about numbers in general?

Bernhard Riemann made a suggestion about how prime numbers were distributed along the number line in 1859. This suggestion became known as the Riemann Hypothesis, and stated that the pattern of distribution of primes is connected to non-trivial zeros of the zeta function. What the hell does that mean? The majority of the book is spent answering that question by unpacking the zeta function.

We have a proof that describes how prime numbers are distributed - it's called the prime number theorem, and Riemann's work was crucial to its development. However, we haven't yet proven the Riemann Hypothesis, which is a much stronger statement than PNT.

Compared to other math books, Derbyshire does a great job of getting into the meaty details and walking through them in a clear manner. The tone was very conversational, doesn't require much linear algebra or calculus, and stuck to the topic at hand (he would often hint at enormous tangents while declaring that he will be avoiding them for our own sake).

I have one main concern, and is the reason why I didn't give this more stars. While I can appreciate the mystery of prime numbers, and I could follow all of the book's content(after much concentration on my part), I had trouble grasping the importance of the Riemann Hypothesis after Prime Number Theorem was proven. However, I suspect that at some point he did explain it, and I since forgot it. For its density, this is a pretty long book (about 400 pages, including the notes, which I read). I'll need to, at some point, return to the book to re-learn the significance of RH from the detailed explanation of its composition.

At the end of the day, RH doesn't excite me as much as the continuum hypothesis or the incompleteness theorems. I know full well this is because RH is far more obscure than, say, the continuum hypothesis or the incompleteness theorems in relation to my pedestrian ontological/epistemological interests. I guess that's why I'm not a mathematician.

If you're interested in this specific topic matter, pick it up. If you just have a pedestrian interest in mathematics like I do, this book is interesting, fun and clear, but there are probably more engaging topics out there than the Riemann Hypothesis.(less)

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May 02, 2009 trivialchemy rated it 3 of 5 stars

This comparison will probably strike most as directly from left field, but Derbyshire reminds me a lot of Jon Krakauer. Topically, of course, they have nothing in common. But their style both depends heavily on the conspicuousness of the author in the narrative. This isn't necessarily because Krakauer and Derbyshire are narcissistic or self-absorbed, but that their writing is very self-conscious and they feel a continual impetus to advise the reader of where they stand on the issues they are pre...more

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Jan 15, 2012 Doug rated it 4 of 5 stars

Very well written, both in terms of the history and in terms of the math. About every other chapter is fairly hardcore mathematics, proofs and integrals and whatnot. The author warns you that you'll need at least freshman year calculus to really get the meat of RH and how approaches to solving it have morphed and changed as the culture of math research has changed.

Mixed in with a straight history of the Riemann Hypothesis and its soldiers is an accidental anthropological analysis of mathematics...more

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Jan 29, 2013 Katia Nosenko rated it 5 of 5 stars

Wonderful book for those who are interested in the subject and the modern math in general. The author has made his task very challengeable - to explain high level abstract math to a layman almost without even using calculus. And i have to say he succeeded! On the top of knowing the subject he is great storyteller. I wish I would have such a teacher for the Calculus when i was at the University;-)

It is not for everyone, but of you are seriously fascinated with math and have some knowledge in the...more

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Jan 08, 2013 Andrij Zip rated it 4 of 5 stars

Shelves: math, history

Prime Obsession is an engrossing and mind stretching journey to the heart of one of the most enduring and profound mysteries in mathematics - the Riemann Hypothesis:

All non-trivial zeros of the zeta function have real part one-half.

By the time you finish the book, that enigmatic statement along with the math behind it will make sense,you will have a deep understanding of the significance of TRH (namely how it is connected to the distribution of prime numbers) and you will have a feel for the ric...more

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Feb 05, 2012 Matt Comstock rated it 4 of 5 stars

Ever wonder what mathematicians do when they're doing research, what they do for a PhD thesis. Wonder no more. This is an introduction to a very complicated area of math. Derbyshire interweaves history and math into an great story. I got about half way through and had to stop trying to completely understand the math he was showing us, and just try to catch the high points. Numbers are very very weird.

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Apr 11, 2013 Eddy Allen rated it 4 of 5 stars

Shelves: arts-and-historical

In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed...more

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Nov 06, 2011 Moonglum rated it 5 of 5 stars

Shelves: pop-math

If you have a bit of a mathematical background, reading this book is like taking a good undergraduate class (but without homework!). It presents a wide range mathematics, all of which is very cool. Really, from the stack of overhanging cards that represents the harmonic series and as such can overhang out over any distance (if you have enough cards), I was delighted.

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Aug 10, 2012 Jake rated it 4 of 5 stars

What an eye opener! Not only did Derbyshire make pre-calc mathematics entertaining but also gave an account to why academics study mathematics. Quirky, witty and downright stimulating and probably one of the best ways to sharpen the mind while learning about a problem that that teased humanity for the past 150 or so years.

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May 23, 2012 M.H. Tardiff rated it 5 of 5 stars

Shelves: non-fiction

John Derbyshire masterfully constructs both an engrossing biography of a lesser-known genius and a first-rate math lesson.

If you have even the slightest interest in math, read this book. You will come away with both a greater appreciation of Herr Reimann and of the cerebral playground made by prime numbers.

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Mar 05, 2012 Scott Lapierre rated it 5 of 5 stars

Totally absorbing. I can't say I fully understand the zeta function, but i at least have a vivid sense of what it signifies, and how it is connected to the real world (or at least the real numbers). For me,the mystery of prime numbers has gone from an unknown unknown to a known unknown.

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Jul 02, 2009 Don rated it 4 of 5 stars

This book alternates chapters -- biographical information interleaved with mathematical explanations. The biographical was fascinating; the mathematics I followed for about six chapters, then fell away exhausted. Overall, a very good read for those interested in such stuff.

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May 24, 2012 Kevin Morgan rated it 5 of 5 stars

I really enjoyed this book, but you have to take your time and work the math (which starts at really easy stuff, and is very carefully explained). You'll never look at prime number, or mathematics for that matter, in the same way again. -kevin aka FitOldDog

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Feb 13, 2009 Karen rated it 3 of 5 stars

Shelves: y-non-fic

What I learned = lots of math.
This wasn't quite what I expected, but I thought the author had a gift for explaining the necessary math (though I'll probably forget it soon enough). The author admits up front that not much is known about Riemann, so it is much less of a biography and much more of a history of the research into Prime Number Theory. I think the "obsession" is on the part of the author, not Riemann.

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Mar 16, 2011 Mike rated it 5 of 5 stars

Shelves: borrowed

An astoundingly good book. Not only is the discussion of zeta fascinating and clear, but we are treated as adults and presented with the actual details of how to calculate relevant values. My thanks to Graham Birtwistle for lending me his copy.

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Aug 20, 2012 Vikash Singh rated it 4 of 5 stars

Shelves: tech

Book clearly explained what is Riemann Hypothesis and different approaches to solve it. Though some chapters contain a bit advanced mathematics (analytic number theory topics) these may arise some confusion.

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Dec 28, 2009 Tim rated it 4 of 5 stars

This was actually quite a good book, covering the life of Bernhard Riemann and courting the reader a glimpse of a mathematical truth that is as yet unknown (if in fact knowable)

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May 26, 2008 Suhaila rated it 5 of 5 stars Recommends it for: Everyone

Recommended to Suhaila by: My dad!

Amazing and mind expanding. It's astounding to think that there are people out there working away at this crazy problem, and how much they have accomplished. I was fascinated by the brilliant leaps of logic and understanding across mathematical disciplines and methods. Yet the Reimann Hypothesis is still unproved.

Derbyshire's writing has piqued my curiosity and interest in both the different fields of mathematics and the principal historical figures of the Reimann Hypothesis. His writing is so...more

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Mar 17, 2009 Michael rated it 5 of 5 stars

The best math book I've ever read. An intriguing combination of history and mathematical exploration. Highly recommended.

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