For a vivid picture of prime fetishism, check out Chris Caldwell's encyclopedic and surprisingly addictive Prime Pages (www.utm.edu/research/primes). Among the contents: a list of the largest known primes (updated weekly), a primality test for numbers smaller than 2^{53} - 1, and abundant prime trivia ("17 was the original title of the Beatles song 'I Saw Her Standing There' "). Over at the Aesthetics of the Prime Sequence (2357.a-tu.net), a counter reels off primes as you listen to spooky minimalist compositions derived by feeding prime numbers into an algorithm, or take a prime intuition test (a crude de facto autism test, perhaps—Oliver Sacks once recounted the bizarre case of autistic twin brothers whose private language consisted of exchanging primes of up to 20 digits).
Prime number theory dates to 350 B.C., when Euclid proved that there are an infinite number of primes. But it was Karl Friedrich Gauss, a future teacher of Riemann's, who in the late 18th century laid the groundwork for the latter's breakthrough. At 14, Gauss observed that while individual primes are randomly dispersed, there is an overall pattern to their density: Not only do they thin out the higher you count, the proportion of primes seems to follow a logarithmic function. (For Gauss, according to Sabbagh's book, counting primes was a pastime—it's said that by the time he died, he'd gotten to 3 million.) His conjecture, that the number of primes less than any given number n is approximately n/log n (proved more than a century later in 1896), is called the Prime Number Theorem.
Gauss's formulation provided only an estimate. Riemann, using the zeta function, derived an exact formula for the number of primes smaller than any given number. The Swiss mathematician Leonhard Euler had discovered that the zeta function, which we defined above as an infinite sum involving all the natural numbers (1, 2, 3, . . . ), could also be expressed as an infinite product involving just the prime numbers. This was, as du Sautoy writes, "the first sign that the zeta function might reveal unexpected links between seemingly disparate parts of the mathematical canon." Riemann's insight was to plug complex numbers into the zeta function—complex numbers have real and imaginary parts, the latter being numbers that can be expressed in terms of i, where i is the square root of -1.
These suspicious-sounding numbers may seem like a flight of mathematical fancy—they were first "imagined" in the 16th century and found to have enormous computational significance, but even mathematicians needed a few hundred years to wrap their heads around the concept, just as it took the ancient Greeks a while to get used to the initially horrifying idea of fractions. (Barry Mazur's recent Imagining Numbers offers an eccentric road map through this abstract terrain.)
Riemann relocated the enigma of prime distribution to the complex plane (a 2-D plane with the x-axis real and the y-axis imaginary). This magical shift, which du Sautoy likens to a beyond-the-looking-glass maneuver, revealed an astonishing regularity. The complex numbers for which the zeta function equals zero are called the Riemann zeros; the frequency of the Riemann zeros mirrors the frequency of the primes. Riemann's famous hypothesis, which he declared sehr wahrscheinlich ("very probable"), was that, mapped on the complex plane, these solutions fall on a straight line—to be exact, the vertical line that runs through the point 1/2 on the real axis.
Convoluted as it sounds, this gloss on the problem grossly simplifies the heady mathematics at its core. All three new books are careful not to outpace the general reader. Du Sautoy, a professor at Oxford, provides a panoramic history of prime-number crunching, rich with anecdote and unfailingly patient with the mathematical fine points. Sabbagh, a documentary producer, uses the R.H. to tunnel into the mathematical mind, folding in copious interviews with number theorists. His approach to the math is at once reassuring and deflationary. Chapter 16 opens: "After fifteen chapters of a book on the Riemann Hypothesis, I have to break some bad news to you. You know almost nothing about the Riemann Hypothesis compared with what there is to know." Derbyshire, a National Review columnist, has written the most mathematically detailed of the trio. Those looking for a quick, lucid R.H. initiation should consult Keith Devlin's evocative chapter in The Millennium Problems (Basic Books, 2002), which also takes on the other six Clay puzzles.
Like many other conjectures, the Riemann Hypothesis could conceivably be disproved by brute tenacity—repeatedly testing it until a counterexample is found. Indeed, the R.H. has been checked—and holds true—for the first 100 billion Riemann zeros. At the AT&T Labs in Princeton, supercomputers keep cranking out zeros, an alarm ready to go off if and when the first zero strays from the so-called critical line.
You'd think this would count as overwhelming evidence in favor of the hypothesis, but mathematics is a science of absolutes, insisting on the concreteness of universal truth as if to compensate for some of its more vaporous abstractions. Du Sautoy says in an e-mail interview, "For the other sciences, evidence in the lab is everything. In the mathematician's lab, such evidence can be completely misleading." In other words, extrapolation is a fool's game when you're dealing with infinity. "The primes are a malicious bunch of numbers," du Sautoy continues. "They're the masters of disguise. People have made conjectures about primes supported by billions of pieces of computer evidence, but theoretical analysis has later revealed the evidence to be a charade."