John Derbyshire's Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics is a wondrous and eye-opening book --- if you've already taken a class in number theory, that is. If you haven't, your faith will be tested repeatedly. Derbyshire is brutally honest when he writes: " ... I shall skip nimbly over the really difficult parts ...", "... I am going to vault over to the following fact ...", "... I can explain this, but only in outline ...", etc.
Yep, as Parappa the Rapper says, "You gotta believe!" Why? The focus of Derbyshire's book, that "Greatest Unsolved Problem" alluded to in the subtitle, is the Riemann Hypothesis, aka the "RH". It dates back to the mid-1800s and has been the focus of a huge amount of work by some of the smartest mathematicians in the world.
What's the RH all about? To paraphrase Richard Feynman's comment, if it were easy to explain then it wouldn't be a great problem --- and Derbyshire's book wouldn't be the tour de force that it is.
Begin with prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. Primes are positive integers that aren't divisible by any other positive integer (except 1). The primes are the building-blocks of huge chunks of mathematics. They're like atoms, important in much the same way that the elements --- hydrogen, helium, lithium, beryllium, boron, carbon, nitrogen, oxygen, ... --- form the foundations of chemistry. Any positive integer can be written as a unique product of primes multiplied together. And there are infinitely many primes.
So primes are important. How are they scattered amongst the other numbers? The sequence looks like a ragged mess. Primes start out appearing close together, but then bigger and bigger gaps show up. Yet every so often, there's another cluster of primes near one another. They seem to be "random" in some ways, but highly patterned in others.
That's where the RH comes in. Derbyshire explains, in beautiful and lucid prose, the mathematical-historical meaning of the RH. As Enrico Bombieri observes in a (highly technical) essay for the Clay Mathematics Institute : "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers. This fact alone singles out the Riemann hypothesis as the main open question of prime number theory." And that's why reading a book like Derbyshire's, and getting a glimpse of the RH, is worth the investment of time for a person who wants to be acquainted with the central questions of our age.
John Derbyshire himself is a fascinating fellow --- articulate columnist for National Review, entertaining and idiosyncratic writer, born in England, resident of New York, married to a lovely Chinese lady, father of some beautiful kids. I find much to disagree with in his commentary as well as much to salute. For instance, Derbyshire muses on life in an interview by Bernard Chapin in Enter Stage Right :
It is highly unlikely that any one of us is a uniquely talented individual with a precious gift to offer the world. It is vastly more probable that we are mere atoms in the mass of humanity, who must find fulfillment in a lifetime of performing humdrum tasks on behalf of our family, neighbors, and fellow-citizens, while we each explore our individuality in small rewarding hobbies and private devotions.
Maybe so. At least, that's how things have been for most folks throughout most of history. But perhaps there has been progress, in the past several millennia, as an increasing fraction of the world has climbed a few steps up from cruelty and illness, ignorance and hunger. "Please, sir, I want some more."
(see also HumanNature (5 Dec 1999), AnalysisAndAlgebra (15 Dec 1999), SeverePrivilege (7 Jun 2000), My Religion (6 Nov 2000), WeHappyFew (11 Mar 2001), ReadingsOnThinkingAndLiving (1 Nov 2001), MillenniumMath (5 Dec 2002), ... )