In a brief mathematical excursion, Bernhard Riemann, then professor of mathematics at Gottingen in Germany, presented a short paper in 1859 on how he thinks prime numbers are distributed among the whole numbers. This became known as the famous Riemann hypothesis – considered one of the great unsolved problems of mathematics.

Prime numbers have long fascinated mathematicians. They are numbers divisible only by themselves and one, like two, three, five, seven, eleven and so on. It’s easy to go on for a while but once you reach the thousands, it becomes a difficult task. Just how many primes are there and is there logic in the way they grow in size?

Thinking of a number as a multiple of another was a construct known to the ancient Greeks for whom a number is either a composite or a prime. Erathosthenes provided a “sieve” whereby any multiple of a number cannot be construed as prime. Hence, all even numbers cannot be prime. Euclid, the master geometrician, showed a procedure for producing larger and larger primes, proving the infinity of primes.

The French mathematician, Adrien-Marie Legendre (1752-1833), whose “method of least squares” has borne significant fruit in astronomy and statistics, with the knowledge that primes veer slightly away from a straight line (i.e., asymptotically) as they grow, proposed a calculating procedure for counting primes using the concept of the logarithm (“The number of primes less than a given value is asymptotically that value, divided by its logarithm”). Carl Friedrich Gauss (1777-1855), the German giant of math, using calculus subsequently developed a Prime Number Theorem. This “logarithmic integral” still holds water now as it comes very close to modern approximations. Then the Swiss mathematician, Leonhard Euler (1707-1783), building on this and Euclid, once more proved the infinity of primes using the concept of the reciprocal and the harmonic series. The reciprocal of 2 is 1/2, of 3. 1/3, and so on. Summing them up yields a harmonic series. Euler “factored” the harmonic series, where each prime contributes, and where each contributing prime can in turn be expressed as an infinite series. Gustav Dirichlet (1805-1859), Riemann’s teacher, did the same using the simpler concept of an arithmetic progression.

Riemann’s lecture on primes was on the occasion of his induction into the prestigious Berlin Academy. He is as well known for his contribution to geometry – the so-called Riemannian geometry which formed the backbone of Einstein’s Theory of General Relativity. A sphere is an example of such a Riemannian space where if you draw a triangle on its surface, the angles add up to greater than the well-known 180 degrees of a flat space. Using the calculus of complex numbers (numbers which consist of a “real” and an “imaginary” part), invented numbers now central to modern physics, he formulated the Riemann zeta function. Think of this function as a box where you put in complex numbers, process them according to some rules, and out comes another number. The process inside the box utilizes ideas from Fourier, another famous French mathematician. The complex number inputs for which the zeta function of Riemann is zero are called zeta zeros. Some of these zeta zeros fall on the “real” axis. They are essentially the negative even integers e.g. -2, -4, -6. Those which do not are called nontrivial zeta zeros. All these nontrivial zeta zeros occur on an imaginary vertical strip corresponding to 1/2 on the real axis. That this is the most precise estimate of the distribution of prime numbers among the natural numbers is the Riemann hypothesis.

The above paragraphs are my bare outline of what constitutes the first half of Dan Rockmore’s exposition of the Riemann hypothesis and its explorers, *Stalking the Riemann Hypothesis* (Pantheon Books, 2005). That’s at least how I understood it. The second half provides the nonmathematical layman a narrative of the the modern attempts to prove the Riemann hypothesis. In the final chapters, the attempts at solving it lead to the frontiers of how we understand the world and the more abstract ideas of quantum chaos.

Rockmore’s book is not for everyone, especially if you hate math or if you’re already a math professional. For the mathematically inclined, however, this is a book to be savored. It is written in an easy-to-like conversational style. Nibble at it, page after page, and enjoy its many flavors. In the end, you’ll be surprised that you understand large chunks of it.

## July 5, 2006

### The Trail Riemann Left Behind

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