Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics: Amazon.co.uk: John Derbyshire, National Academy of Sciences: Books

Amazon.co.uk Review

Bernhard Riemann was an underdog of sorts, a malnourished son of a parson who grew up to discover one of the greatest problems in mathematics. In Prime Obsession, John Derbyshire deals brilliantly with both Riemann's life and that problem, which was to find proof of the conjecture "all non-trivial zeros of the zeta function have real part one-half".

That statement may be nonsense to anyone but a mathematician but Derbyshire walks the reader through the decades of reasoning that led to the Riemann Hypothesis in a way that makes it perfectly clear. Riemann never proved the statement and it remains unsolved to this day.

Prime Obsession offers alternating chapters of step-by-step maths and a history of 19th-century European intellectual life, letting readers take a breather between chunks of well-written information. Derbyshire's style is accessible but not dumbed-down, thorough but not heavy-handed. This is among the best popular treatments of an obscure mathematical idea and allows readers to explore the theory without insisting on page after page of formulae.

In 2000, the Clay Mathematics Institute offered a one-million-dollar prize to anyone who could prove the Riemann Hypothesis, but luminaries like David Hilbert, GH Hardy, Alan Turing, André Weil and Freeman Dyson have all tried before. Will the Riemann Hypothesis ever be proved? "One day we shall know," writes Derbyshire and he makes the effort seem very worthwhile. --Therese Littleton, Amazon.com

Product Description

In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark - a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic - defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark - the Riemann Hypothesis - that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows - subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations.The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many - the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution.Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, "Prime Obsession" is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, "Prime Obsession" is the engrossing tale of a relentless hunt for an elusive proof - and those who have been consumed by it.

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41 of 41 people found the following review helpful

The Greatest Unsolved Mathematical Problem 19 May 2003

By Andrew Hanna

Format:Hardcover

Prime Obsession, is a wonderful book based on the history and insight of the brilliant mathematician, Bernhard Riemann. As the title suggests, the main aim of the book is to give the reader a clear and understandable definition of what the Riemann Hypothesis actually is. To do this, Derbyshire has structured the book so the reader is given a chapter of mathematical tools, followed by a chapter of the history of Riemann and other great mathematicians, such as Gauss, Euler, Hardy, followed by a math chapter etc... However, don't let the math sections put you off this book, as Derbyshire explains, he uses minimal calculus to get the reader through the book. He takes the reader though basic analysis, then onto prime numbers, domain streching, followed by what he calls the Golden Key which uses the Euler product. Then he introduces basic complex number theory, and finally he pulls them all together to start to explain the RH (Riemann Hypothesis). Riemanns ideas and visualizations of complex functions are difficult to comprehend for even the most accomplished mathematician, but Derbyshire employs a method that any lay person can understand perfectly, using his "Argument Ant". Any person interested in mathematics, should read this book, as it serves as a wonderful insight into one of the greatest mathematicians, and problems that has ever existed. And for those who are just interested in the RH but were never quite sure where the zeros come from, then the chapter on domain streching and subsequent chapters will make it all clear. This is the best popular science book I have read since Feynmans "QED: The strange theory of light and matter".

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34 of 34 people found the following review helpful

Do not buy any others 13 April 2007

By Hamptonshirewonder

Format:Paperback

I have read this book and two of the other three popularisations about the Riemann hypothesis. Instead of interviewing mathematicians who may be near to solving it or writing around the subject, this book actually works through the mathematics of Riemann's 1859 paper.

"Prime obsession" emphasises the centrality of the other parts of Riemann's paper apart from the famous Hypothesis. By doing this it helps to explain why some 30 years later that mathematicians were able to prove the Prime Number Theorem, independently of the truth or otherwise of the famous Hypothesis. The Prime Number Theorem states, roughly that: as numbers get larger the number of primes less than that number tends to about the number divided by its logarithm (base e). The reason the Prime Number Theorem could be proved, irrespective of Riemann's Hypothesis' truth, is because of the techniques that Riemann invented in his 1859 paper.

Riemann's starting point was to generalise Euler's formula which relates the sum of the reciprocals of natural numbers:

1+1/2+1/3+1/4+...

to the product of the inverses of the prime numbers

(1/2)*(1/3)*(1/5)*(1/7)*(1/11)*.....

Derbyshire's explanation is far clearer and much easier to follow than those in the other popularisations.

This book is precise and clear: one really feels that one has some insight into an astonishing piece of creative mathematical work by the time one has read the book. That alone in my opinion should qualify it as one of the greatest pieces of popular science writing of this or any other decade.

This book needs to be more actively marketed: whatever its faults, the author has made a genuine attempt to really explain a great piece of science technically to a non -technical audience, rather than just waffling around the subject and making us all feel these things are so far above our heads we will never understand them in any way. This courage on the author's part needs to be more widely feted.

I cannot do more than endorse the other reviewers' praise for this classic-to-be. for those interested in pursuing this fascinating subject further, I found Gamma: Exploring Euler's Constant (Princeton Science Library) by Havil to be a wonderful book.

15 of 15 people found the following review helpful

A class of its own 18 Dec 2008

By Simon Pollack

I am a bit of a junkie for books on maths, revisiting my degree of 15-20 years ago. The quality varies a lot though and I am very often disappointed. This I supose is not surprising: I want not to be patronised but I also want accessibility, context (historical, personal), and some insight into the underlying beauty of the mathematics in question. But this book pushes all the right buttons.

The Riemann Hypothesis is really quite advanced - you wouldn't find much in-depth study of it in any compulsory modules of undergraduate courses. But Derbyshire brings it to life. The book is challenging but accessible, and ultimately a very fulfilling read.

I think the key to his success is the interleaving of chapters on the lives of the protagonists with those on the maths leading up to and surrounding the Hypothesis. Because an understanding of the relevant mathematics helps understand the importance of a given mathematician's life, and an understanding of historical context helps bring the maths to life, these chapters are mutually reinforcing. As such the whole is greater than the sum of the parts (I think I might just have found that 1+1>2). And because so many of the great mathematicians contributed to the foundations of number theory and analysis, and many subsequently worked on the Riemann Hypothesis itself, this book kind of doubles as a selective history of modern (from Newton) mathematics.

I can't recommend this book enough. Even for those with no background in maths, but with an enquiring spirit, there is enough here (crucially, without turgidity) to dimly comprehend the profound beauty and true mystery of maths. It makes you believe somehow in the Platonic Ideals and that those blessed with true insight get closer to them than the rest of us. I have always felt that advanced pure mathematics is as worthy an art as painting or sculpture, and the great mathematicians as worthy artists as Van Gogh etc. But because of the inaccessibility of the subject matter to the layman this great art couldn't be widely-enough shared. With more books like Prime Obsession this wrong will be righted.

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