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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
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.
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.
340 pages, Hardcover
First published January 1, 2002
17 people are currently reading
627 people want to read
About the author
Karl Sabbagh
26 books22 followersKarl Sabbagh, founder and managing director of Skyscraper Publications, has written nearly a dozen books, ranging across topics as diverse as architecture, psychology, history, mathematics, fraud, Victorian boys’ papers, and the Middle East. Some of his books are derived from major television documentary series he produced and directed; others are pieces of original non-fiction for a general readership.
From 2010 to 2012, he was managing director of Hesperus Press, an independent British publisher of minor classics, fiction in translation, and some original non-fiction. While at Hesperus, he acquired the UK rights to The Hundred-Year-Old Man who Climbed Out of the Window and Disappeared, which has so far sold over 500,000 copies in the UK alone.
Skyscraper's unique programme draws on Karl's extensive experience as both author and publisher.
From 2010 to 2012, he was managing director of Hesperus Press, an independent British publisher of minor classics, fiction in translation, and some original non-fiction. While at Hesperus, he acquired the UK rights to The Hundred-Year-Old Man who Climbed Out of the Window and Disappeared, which has so far sold over 500,000 copies in the UK alone.
Skyscraper's unique programme draws on Karl's extensive experience as both author and publisher.
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Displaying 1 - 29 of 29 reviews
December 8, 2011
The Riemann Hypothesis is easily one of the most important unsolved problems in mathematics. Unfortunately, unlike Fermat's Last Theorem or even the Poincaré conjecture, the Riemann Hypothesis doesn't lend itself well to being put in layman's terms. Even still, Sabbagh gives quite the valiant effort in his book, and in the process takes us into the strange world of cutting-edge mathematics.
Sabbagh's trip through the Riemann Hypothesis is a slow, methodical one: he starts out with some required fundamentals: prime numbers, imaginary numbers, complex numbers, and works his way up to more esoteric topics like infinite series, zeta functions, and finally what Riemann's Hypothesis says about the zeros of these functions. Along the way, Sabbagh introduces various mathematicians whom, although you've probably never heard them, are larger-than-life in their own right, and what life is like for these people who concentrate intensely on very abstract concepts.
As I've mentioned in other reviews, I love books that talk about the 'romantic' side of mathematics, and Sabbagh does a wonderful job of that in his book. Many of the mathematicians he interviewed I had never heard of before, but their stories were so enthralling that I never once felt a twinge of boredom.
My only real criticisms of the book lie in it's flow. I know that trying to explain the Riemann Hypothesis in non-mathematical language is a very daunting task, but the explanations and analogies scattered throughout the book didn't seem to flow well enough to paint an overall picture for me. Also, in some areas, Sabbagh seemed to diverge quite a bit from the main topic of the book. Although the segues themselves are interesting topics, again I felt it disrupted the overall flow.
If you're like me, and you followed the stories behind Fermat's Last Theorem and the Poincare conjecture, I definitely recommend picking up Sabbagh's book and discovering the weird and fascinating world of Riemann's Hypothesis.
Sabbagh's trip through the Riemann Hypothesis is a slow, methodical one: he starts out with some required fundamentals: prime numbers, imaginary numbers, complex numbers, and works his way up to more esoteric topics like infinite series, zeta functions, and finally what Riemann's Hypothesis says about the zeros of these functions. Along the way, Sabbagh introduces various mathematicians whom, although you've probably never heard them, are larger-than-life in their own right, and what life is like for these people who concentrate intensely on very abstract concepts.
As I've mentioned in other reviews, I love books that talk about the 'romantic' side of mathematics, and Sabbagh does a wonderful job of that in his book. Many of the mathematicians he interviewed I had never heard of before, but their stories were so enthralling that I never once felt a twinge of boredom.
My only real criticisms of the book lie in it's flow. I know that trying to explain the Riemann Hypothesis in non-mathematical language is a very daunting task, but the explanations and analogies scattered throughout the book didn't seem to flow well enough to paint an overall picture for me. Also, in some areas, Sabbagh seemed to diverge quite a bit from the main topic of the book. Although the segues themselves are interesting topics, again I felt it disrupted the overall flow.
If you're like me, and you followed the stories behind Fermat's Last Theorem and the Poincare conjecture, I definitely recommend picking up Sabbagh's book and discovering the weird and fascinating world of Riemann's Hypothesis.
Read
May 3, 2023This book doesn’t exactly teach you mathematics, but it pretends to, which is just as good. (There are “toolkits“ in the back, to remind you what logarithms are, etc.)
Did mathematicians sleep with each other’s wives? And if so, do they do it just to illustrate the commutative principal?
This book doesn’t quite reveal that, but there is a lovely section on jokes mathematicians tell:
2 plus 2 equals five for sufficiently large values of 2.
Writing poetry is a lot like writing mathematical proofs. T. S. Eliot’s (failed) attempt to “revive the verse drama” was like these fools trying to prove the Riemann Hypothesis.
Opening at random:
'The popular idea of mathematics is that it is largely concerned with calculations. What many people don’t realize – and mathematicians at parties have given up correcting them – is that mathematicians are often no better calculators, and sometimes worse, than the average nonmathematician. An incident during my first meeting with the Franco-American mathematician Louis de Branges illustrates that nicely. We were discussing the idea that mathematicians did all their best work when they were young, and I asked him when he had some particular insight.
'“Let’s see,” he said. “It happened in 1984 and I was born in 1932. So was I over 50? How old was I then…?”
'He thought for a while, wrestling with the problem as if it were the Riemann Hypothesis itself, and then gave up…'
Did mathematicians sleep with each other’s wives? And if so, do they do it just to illustrate the commutative principal?
This book doesn’t quite reveal that, but there is a lovely section on jokes mathematicians tell:
2 plus 2 equals five for sufficiently large values of 2.
Writing poetry is a lot like writing mathematical proofs. T. S. Eliot’s (failed) attempt to “revive the verse drama” was like these fools trying to prove the Riemann Hypothesis.
Opening at random:
'The popular idea of mathematics is that it is largely concerned with calculations. What many people don’t realize – and mathematicians at parties have given up correcting them – is that mathematicians are often no better calculators, and sometimes worse, than the average nonmathematician. An incident during my first meeting with the Franco-American mathematician Louis de Branges illustrates that nicely. We were discussing the idea that mathematicians did all their best work when they were young, and I asked him when he had some particular insight.
'“Let’s see,” he said. “It happened in 1984 and I was born in 1932. So was I over 50? How old was I then…?”
'He thought for a while, wrestling with the problem as if it were the Riemann Hypothesis itself, and then gave up…'
October 23, 2024
I was expecting a lot more maths to be in the book; it really just went into the different people trying to solve it. However, it was quite interesting to learn about the most influential mathematicians in the field.
February 19, 2020
Sabbagh makes an admirable attempt, and has a clear and sometimes entertaining style, but I was looking for a wholly different book than what this seemed to be.
There is far less actual discussion of the Riemann Hypothesis in this than there are anecdotes about, and sometimes interviews with, some slightly strange mathematicians about things other than their mathematics. The author seems to labour under the common but still disconcerting belief that a mathematician is a wholly different beast than the average human being, and it does come through-- in both the anecdotes and the total avoidance of formulae throughout.
That last bit is particularly strange, considering it's a book about a formula.
There is far less actual discussion of the Riemann Hypothesis in this than there are anecdotes about, and sometimes interviews with, some slightly strange mathematicians about things other than their mathematics. The author seems to labour under the common but still disconcerting belief that a mathematician is a wholly different beast than the average human being, and it does come through-- in both the anecdotes and the total avoidance of formulae throughout.
That last bit is particularly strange, considering it's a book about a formula.
May 22, 2013
I enjoyed this book back when I read it. It gave a nice history of Dr. Riemann, and a reasonable overview of the hypothesis. At one point I think I even almost grasped the whole of it, though it slipped away almost as soon as it had appeared. So I still don't really understand the Riemann Hypothesis, at least not in any deep way. I guess that's because reading popular science isn't a substitute for taking several courses in advanced number theory :-). One day, perhaps. The primes /are/ fascinating.
August 18, 2008
This book sparked my interest in books about math history, mathematicians, and numerology. it makes clear descriptions of abstract concepts, and glimpses into the psyches of some of the most interesting mathematicians in the world.
afterwards i moved onto books on Fermat, Newton, the concept of Zero, and famous diagrammatical proofs.
i just love two-dimensional illustrations of three-dimensional concepts like infinity.
afterwards i moved onto books on Fermat, Newton, the concept of Zero, and famous diagrammatical proofs.
i just love two-dimensional illustrations of three-dimensional concepts like infinity.
November 16, 2012
An interesting book with regard to some of the personalities involved, but it didn't have a great deal of insight into the mathematics. The first three chapters were most interesting and the remainder were fairly useless due to lack of mathematical sophistication.
February 8, 2010
This book was more about the author than the topic. Some of the material is good, but the book minus the author talking about himself is about half as long.
April 28, 2008
The least math-like math book ever written. Thoroughly enjoyable for anybody.
December 29, 2025
Overall too much bio, not enough explaining, such that the book is mostly about who and vaguely what, and very, very little how and why. Around the fourth time we hit material on Louis de Branges, I started counting pages until the end. And thereafter the book does devolve into a downward spiral of people and concept name-dropping with less and less explanations of the concepts that interested the people.
The author excuses himself at the end of the book by saying that it’s impossible to truly tell you what the Riemann Hypothesis is really all about because the math is too difficult. However, the author seemed to go out of his way to not explain anything that gets you anywhere near an approximation of an understanding. The lone exception was the coverage of the Euler identity relating sums of fractions involving all numbers with products of fractions involving primes. Much more of the book needed to be like that. For example, the discussion of eigenvalues and eigenvectors was especially disappointing, owing partly to the fact that here is where Sabbagh could have gotten the reader closest to having a true flavor and aroma of the Riemann Hypothesis. And that Sabbagh could have done so without breaking everyone’s brains is evident now because these days you can do a web search for “why are eigenvalues and eigenvectors important “ and land on very nice math stack exchange posts that can be accessed for free<\i> because people love this stuff and want to share that passion with others.
I would have really liked for this book’s author to have more of that kind of passion for the subject matter. Failing that, my recommendation about this book can be expressed as a corollary of the sharp-witted “endorsement” by Scientific American writer Kristin Leutwyler on the back of my copy of the book: “The Riemann Hypothesis provides ample hand-holding for anyone who pales at the sight of symbols or can’t quite distinguish an asymptote from a whole in the graph.” The corollary is that if you don’t pale and you can distinguish, then you should undoubtedly pick a different book.
The author excuses himself at the end of the book by saying that it’s impossible to truly tell you what the Riemann Hypothesis is really all about because the math is too difficult. However, the author seemed to go out of his way to not explain anything that gets you anywhere near an approximation of an understanding. The lone exception was the coverage of the Euler identity relating sums of fractions involving all numbers with products of fractions involving primes. Much more of the book needed to be like that. For example, the discussion of eigenvalues and eigenvectors was especially disappointing, owing partly to the fact that here is where Sabbagh could have gotten the reader closest to having a true flavor and aroma of the Riemann Hypothesis. And that Sabbagh could have done so without breaking everyone’s brains is evident now because these days you can do a web search for “why are eigenvalues and eigenvectors important “ and land on very nice math stack exchange posts that can be accessed for free<\i> because people love this stuff and want to share that passion with others.
I would have really liked for this book’s author to have more of that kind of passion for the subject matter. Failing that, my recommendation about this book can be expressed as a corollary of the sharp-witted “endorsement” by Scientific American writer Kristin Leutwyler on the back of my copy of the book: “The Riemann Hypothesis provides ample hand-holding for anyone who pales at the sight of symbols or can’t quite distinguish an asymptote from a whole in the graph.” The corollary is that if you don’t pale and you can distinguish, then you should undoubtedly pick a different book.
January 18, 2021
The book is okay as math popularizations go. I sometimes turn to youtube. If you want to get a handle on what's involved in the Riemann Hypothesis and the Riemann Zeta function and its relation to the distribution of primes and why there is a cool million waiting for the person who proves the Riemann hypothesis I will drop a video or two below.
https://www.youtube.com/watch?v=sD0Nj...
https://www.youtube.com/watch?v=d6c6u...
https://www.youtube.com/watch?v=YuIIj...
https://www.youtube.com/watch?v=sD0Nj...
https://www.youtube.com/watch?v=d6c6u...
https://www.youtube.com/watch?v=YuIIj...
January 12, 2022
I finished reading this and then looked up the author and it turns out he's a convicted paedophile. Uh-oh. Oh well.
I actually got this a long time ago when I was interested in doing a maths degree. That didn't work out in the end (long story short) but I kept the book because I found this subject interesting from a layman's perspective. I think the author did a good job of explaining it in layman's terms. But he seems a bit enamoured by this one mathematician that the maths community doesn't take seriously... and then prints an excerpt of the supposed proof that this guy came up with in an appendix at the end of the book, which is impenetrable to a layman. So there's a bit of a jump there between knowing nothing and knowing everything that he doesn't really bridge properly.
I actually got this a long time ago when I was interested in doing a maths degree. That didn't work out in the end (long story short) but I kept the book because I found this subject interesting from a layman's perspective. I think the author did a good job of explaining it in layman's terms. But he seems a bit enamoured by this one mathematician that the maths community doesn't take seriously... and then prints an excerpt of the supposed proof that this guy came up with in an appendix at the end of the book, which is impenetrable to a layman. So there's a bit of a jump there between knowing nothing and knowing everything that he doesn't really bridge properly.
November 24, 2024
It's a well researched and well organized book. But I don't really know who this book is for. It's too mathy for most people to take interest, and not mathy enough for the mathy people (me) who ARE interested.
March 11, 2018
Well written but quite hard going. I enjoyed it however.
Read
August 9, 2019A wonderful look at the Riemann Hypothesis, and one of the books that sparked my excitement for a career in mathematics.
April 16, 2021
This is still not proved but it looks like Louis is still trying
This entire review has been hidden because of spoilers.
August 20, 2022
A brilliant and readable treatment of this fascinating mathematical enigma. I've recommended this to many of my math students to pique their interest in higher-level math.
November 10, 2025
This book is less of a technical guide to one of mathematics’ greatest mysteries and more of a journey through the people passions, and peculiarities that have surrounded it for over a century. Sabbagh writes with clarity and warmth, making the subject accessible and the mathematics approachable.
That said, if you came here hoping to understand the Riemann Hypothesis itself, prepare to leave with about the same level of comprehension you entered with—just with a lot more opinions about 19th century mathematicians. It’s definitely a prime way to feel smarter than you really are.
That said, if you came here hoping to understand the Riemann Hypothesis itself, prepare to leave with about the same level of comprehension you entered with—just with a lot more opinions about 19th century mathematicians. It’s definitely a prime way to feel smarter than you really are.
August 3, 2010
(disclaimer- alcohol was consumed during the creation of this review, ideas may not be logically coherent).
is very well known for his hypothesis. This book about Georg Friedrich Bernhard Riemann has taught me that the behaviour of the zeta function is pretty serious business. This is a zeta function: it
-> ζ(s) <-
where s is a complex variable. If the real part of this variable is greater than one then ζ(s) is defined as the sum of the convergent series ∑n ≥ 1 n^-s (a convergent series is converging, which is the arch-nemesis of diverging). Oh-and you have to extend ζ(s) through analytic continuation so that it may inhabit the whole complex plane (as opposed to simple plane, like a Fokker Dr.1). When you mash all this together, Riemann says that the real part of this complex number s is exactly 1.5 when it is between 0 and 1, and ζ(s) = 0. When this idea first showed up in Riemann's work, there were no clue as to how he developed it.
Riemann's conjecture that all of the zeroes of the zeta function have a real part of 1.5 showed no proof whatsoever. which could imply that Riemann was from outer space because his math was too complicated for our monkey brains to grasp. He teleported away to his advanced civilization where they watch sitcoms about earth-people and lol all day.
(to be continued)
is very well known for his hypothesis. This book about Georg Friedrich Bernhard Riemann has taught me that the behaviour of the zeta function is pretty serious business. This is a zeta function: it
-> ζ(s) <-
where s is a complex variable. If the real part of this variable is greater than one then ζ(s) is defined as the sum of the convergent series ∑n ≥ 1 n^-s (a convergent series is converging, which is the arch-nemesis of diverging). Oh-and you have to extend ζ(s) through analytic continuation so that it may inhabit the whole complex plane (as opposed to simple plane, like a Fokker Dr.1). When you mash all this together, Riemann says that the real part of this complex number s is exactly 1.5 when it is between 0 and 1, and ζ(s) = 0. When this idea first showed up in Riemann's work, there were no clue as to how he developed it.
Riemann's conjecture that all of the zeroes of the zeta function have a real part of 1.5 showed no proof whatsoever. which could imply that Riemann was from outer space because his math was too complicated for our monkey brains to grasp. He teleported away to his advanced civilization where they watch sitcoms about earth-people and lol all day.
(to be continued)
November 10, 2008
The funny thing about this book, for me, is that it claims to simply be a study of how the Riemann Hypothesis has been scrutinized, fought over and beloved by mathematicians. It is not supposed to be analysis of the theory itself. And yet for me, a non-mathematician, it has gotten me the closest to understanding the R.H.
And the anecdotes are simply great.
And the anecdotes are simply great.
January 10, 2017
Karl Sabbagh makes a fair attempt at unveiling some of the mystery behind the illustrious Riemann Hypothesis. Doing so by walking the reader through the lives of mathematicians who've dedicated their careers, and in some cases their lives, towards solving this problem, while illustrating in layman's terms their attempts at breaking down such a simple yet vastly complex problem.
April 20, 2025
I enjoyed learning about the such an import piece of mathematics. The Riemann hypothesis is connected to the distribution of prime numbers. I definitely feel that it takes more reading to get a handle (if you're mathematically inclined) on the mathematics involved. However, it's a great overview of Riemann and the Zeta function.
August 15, 2016
Interesting to learn more about the mathematicians working with the Rieman Hypothesis. I wished to learn more about the actual theorem but this book is more of a collection of interviews and reflections around the interviews.
July 2, 2011
A fun book about a subject that the author could not quite get me excited about. But, he was a good story teller.
November 26, 2012
OK, especially if you don't know much about math (although you might then be unlikely to read the book!), but Derbyshire's book is a lot more informative (though a tougher read).
August 3, 2013
Although written for the layman, you'd have to be pretty mathy or wanna be mathy (like me) to read this for enjoyment. I liked it quite a bit.
March 27, 2014
Could have done with a little more mathematics
July 3, 2024
p14: "There are 92 naturally occurring elements, from hydrogen to plutonium..."
Ummm.
Read John Derbyshire's "Prime Obsession" instead.
Ummm.
Read John Derbyshire's "Prime Obsession" instead.
February 25, 2009
A challenge unless one has a solid math background.
Displaying 1 - 29 of 29 reviews