03 April 2007

Riemann hypothesis

My recent reading has included a book by John Derbyshire, who bravely attempts to explain the Riemann hypothesis, and efforts to prove or falsify it, to those of us without much formal education in math.

This problem seems to have been the inspiration for the drama Proof, which won the 2001 Tony Award for best play and became the basis for a Hollywood movie. That movie starred Gwyneth Paltrow as Catherine, a young woman who learned mathematics from her father, and who may have devised an extraordinary solution to a long-standing puzzle about prime numbers. [On Broadway, Mary-Louise Parker was Catherine.]

In the meantime, and in the real world, the Riemann hypothesis remains unproven.

What is it? Just to test whether I've been wasting my time with the book, I'll try to explain it now. It involves the distribution of prime numbers, and the fact that primes are bunched up in the lower numbers but become more scarce further up the line. The early primes are: 1, 2, 3, 5, 7, 11, and so forth. They're already beginning to thin out, even in that brief series.

The thinning continues. There are 26 primes of less than 100. Between 401 (inclusive!) and 500, there are just 17. Between 901 and 1,000, there are only 14 primes.

Is there a rule that governs this thinning out? Yes, the rule is (unsurprisingly) called the Prime Number Theorem (PNT). That isn't the Riemann hypothesis (RH) though. The RH proper is something Riemann suggested as a rather incidental spin-off of his work on establishing the PNT.

In order to state the PNT, one has to invoke complex numbers (numbers with both a real and an imaginary component), and the complex value of something Riemann called the zeta function, because he was too humble to call it the "Riemann zeta function" as everyone else now does, http://en.wikipedia.org/wiki/Riemann_zeta_function

The Riemann hypothesis is that the real part of any non-trivial zero of the zeta function is 1/2.

We're almost to the point where that bare statement makes some sense. The "zero" of any function is that value at which the function produces a result of zero. Simple example: x squared – 6x + 9 = y. [I write "x squared" because I haven't figured out superscripts here.] The "zero of f(x)" in the relevant sense is that value of x for which y = 0. In this case, 9 - 18 + 9 = 0, so the zero of f(x) is 3.

The zeta function generated by the search for the PNT though, turned out to have an infinite number of zeros, each complex numbers. Some of these zeros (the "trivial" ones) are generated when the variable s (don't ask) is an even negative integer: -2, -4, -6 and so on. But that leaves us with the non-trivial zeros and the question: do they all (as Riemann believed) have a real component of exactly 1/2?

No wonder the Anthony Hopkins character lost his mind, and Catherine seemed on the way to losing hers.

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