Review of Prime Obsession

By Greg Mone

Prime Obsession, the story of what many mathematicians consider to be the great unsolved problem of their field, begins as two very different books.  The first details the life of Bernhard Riemann, the 19th century German mathematician who influenced everything from number theory to general relativity.  The second is a guided tour through his most vexing contribution, the Riemann hypothesis.  As you ascend the number scale, the frequency of primes — those numbers divisible by only one and themselves — drops off in a pattern that Riemann described but never proved.  Author John Derbyshire, both a novelist and a mathematician, writes in the prologue that he originally intended to divide this book in two.  Less math-inclined readers could choose to read only the chapters that covered Riemann's interesting life and skip the dreaded symbols, numbers and graphs required to explain his famous hypothesis.  In its final form, however, the two narratives have merged.  As Riemann's story progresses and the shy prodigy matures into a devoted mathematician, the mathematics becomes his life.  The Riemann hypothesis, as Derbyshire shows through approachable examples and colorful quotes from leading mathematicians, has now acquired a life of its own.  It is hardly easy to explain, but Derbyshire does his very best. He also takes his time to do so.  We don't learn exactly what the Riemann Hypothesis is until we're well into the second half.  In short, Riemann developed a complex function that reveals a pattern to the distribution of primes, but he didn't provide a proof.  For over one hundred and fifty years, mathematicians have been trying to finish the job. There have been advances, retreats, false victories declared and hollow predictions revealed.  Through it all, the Riemann hypothesis remains, as daunting as ever.

For British science writer Karl Sabbagh, whose book, The Riemann Hypothesis, is also being published in the U.S. this month, the challenge of the proof and the difficulty of the mathematics involved produced a very different work from Derbyshire's.  According to Sabbagh, the real story is the mathematicians working on the hypothesis, and he lets them speak for themselves.  (He even walks us through a few complex but apparently hilarious math jokes that few readers will get.)  You need only flip the pages of the two books to see that they're very different:  The Riemann Hypothesis is mostly dialogue, culled from interviews with the world's leading mathematicians.  Where Derbyshire tries to teach the reader to understand and appreciate the immensity of the problem as a mathematician would, Sabbagh lets the characters try to convince you on their own.  The truth is that mathematicians are, in their own way, funny.  They're strange, unusual, and, sometimes, oddly normal.  Some joke about the inapplicability of their work to the real world, others point out that it's not the solving of these problems that produces useful things, but often the tools they develop to do so.

As both books make clear, the RH, as it's called, had become an obsession for many a mathematician long before May, 2000, when the Clay Mathematics Institute, founded by Boston-area financier Langdon T. Clay, announced an award of $1 million to anyone who could provide a proof.  To Derbyshire, this amount is little compared to the fame and possible fortune that would come to the RH's eventual conqueror.  In fact, he hints that $1 million is too little.  The Clay Institute also announced million-dollar grants for six other unsolved problems and the others, according to Derbyshire, are not as daunting.  David Ellwood, a mathematician at the Clay Mathematics Institute, said in a recent interview that these seven problems are not necessarily the most fashionable in the field.  "Mathematics is evolving and there are many new interesting mathematical problems," Ellwood said.  "But we wanted to draw attention to classical unsolved problems."  And the Riemann, he concedes, is the foremost among them.  "The others on the list are comparable to the RH, but don't carry the same cachet."

At the close of his difficult but rewarding book, Derbyshire concludes that this proof remains a long way off. Ellwood agrees: "It's so difficult that it's not something we feel we can solve right now."