And You Thought Prime Numbers Were Dull

kw: book reviews, prime numbers, Riemann hypothesis

Am I a glutton for punishment, or what? My career is on the fringes of mathematics, mainly algorithmics. I have a dilettante's interest in quite a variety of mathematical disciplines. But those weird critters called "higher mathematics" are so far beyond my ken...

Well, I found myself reading almost compulsively, as I made my way through Stalking the Riemann Hypothesis by Dan Rockmore, published by Pantheon Books. I didn't count them up, but I think Dr. Rockmore touches at least fifty realms of mathematics with which I have little or no experience. He shows how all of them build on one another, how all are linked in various (and to me, abstruse) ways, and how all—and others—will be needed to prove the Riemann Hypothesis.

Rockmore explains just enough of number theory, the hierarchy of numbers, logarithms, calculus, topology, and a host—a big host—of other aspects of math so a moderately educated reader can at least follow where he is going. Not many will really understand, but it is like following a guide who can tell you more detail about every flower, bug, waterfall, and rock than you'd ever absorb. You pick up enough to enjoy the landscape a bit better than before.

He also presents capsule biographies of a few dozen people who have contributed to the understanding of the Riemann hypothesis, the connection between his "zeta zeroes" and the "zeta zeroes" of many, many other mathematical systems. He is a great example of a scientist who not only writes well, but can tell a good story.

One might say this book is premature. The hypothesis hasn't been proved, so...what's the big deal? For one thing, I realized that, if the hypothesis is ever proved, at least a (very) few people will have a deep and clear understanding of the deep structure hidden in the endless parade of prime numbers.

It all centers on efforts through several centuries to improve, and prove, the Prime Number Theorem (Here is the Wikipedia discussion). Very basically, the number of primes less than n is estimated asymptotically to be n/ln(n). For n = 100, this gives 21.7 → 22. There are actually 25 primes smaller than 100. For n = 1016, n/ln(n) yields 271,434,051,189,532 and change; the actual number is 279,238,341,033,925. The discrepancy is 3%.

Gauss took the approach of integrating this function, which produces much greater accuracy, in the range of 0.01% or less. Riemann used complex analysis to produce a function with an infinite number of zeroes, or roots: values for which the function produces zero. A major, infinite, family of these zeroes lie on a vertical line in the complex plane, with a real component of 0.5; to prove the Riemann Hypothesis amounts to proving that all of the zeroes of interest fall on this line.

Appropriate computations with this function reproduce the prime accumulation curve very exactly. An example in the book shows how well it works for primes less than 230. The kicker, IMHO, is that the close correspondence shown required doing computations using the first 500 of Riemann's zeta zeroes. If this trend holds, it seems to me that actually finding primes by using Riemann's function requires much more computation than any way in which we do it now

However, the value in finding a proof is that the existence of the proof would show that there is an underlying order in the sequence of prime numbers, beyond the obvious one embodied in the sieve of Eratosthenes: this explanation at The Prime Glossary is better than I can devise. Basic take-away message: the farther you take "the sieve", the more prime divisors are needed, so the ranks of primes get thinner and thinner as more groups of multiples are removed.

I give Dr. Rockmore great credit for understanding the many corners of the math landscape well enough to make them, if not fully understood, certainly quite satisfying conceptually and enjoyable to contemplate.