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August 23, 2007

THE INDIAN CLERK: FURTHER READING # 3

Pin2 If you're of a mind to delve more deeply into the maths of The Indian Clerk, you do well to read one of these four volumes devoted to The Riemann Hypothesis, and recommended by Leavitt himself, in his order of preference:

1. Marcus du Sautoy, The Music of the Primes.  You can read a fine article by du Sautoy in Plus Magazine, which serves as a fine introduction to Riemann.

Riemann had found one very special imaginary landscape, generated by something called the zeta function, which he discovered held the secret to prime numbers. In particular, the points at sea-level in the landscape could be used to produce these special harmonic waves which changed Gauss's graph into the genuine staircase of the primes. Riemann used the coordinates of each point at sea-level to create one of the prime number harmonics. The frequency of each harmonic was determined by how far north the corresponding point at sea-level was, and how loud each harmonic sounded was determined by the east-west frequency.

2. Dan Rockmore, Stalking the Riemann Hypothesis.  Rockmore and his book were favorably noted by The Washington Post, which observed that "math is hot."

To understand how someone can spend hours, days, years wrestling with an insoluble problem, you have to look at the world through a mathematician's eyes. That's where Rockmore comes in. He's not one of those fluky-flakey number nerds you read about. He's a hiker, a tennis player, a distance runner. He's got a loving family, a Manhattan pied-à-terre and patience enough to explain math to the unmathematical. He is an expositor who scored higher on his verbal SATs than on his math and he has agreed to spend the afternoon walking you through some of the toughest concepts in math -- literally.

3. Karl Sabbach, The Riemann Hypothesis.  About which we can find next to nothing.

4. John Derbyshire, Prime Obsession.  You can sort through a variety of reviews, including this one from the American Journal of Physics.

Presented in a conversational style, but with the meticulous attention to detail of a well-composed detective novel, Prime Obsession tells of the origin, evolution, and significance of a mathematical conjecture with deep ramifications throughout many fields of mathematics and surprising physical implications still to be explored fully. Seamlessly the author weaves together the "world lines" of Riemann and the eminent mathematicians who either motivated or followed up on his work, explaining carefully and readably the essential mathematical contributions made by each.

We were hoping to dig up a reasonably straightforward description of the Riemann Hypothesis but as Andrew Wiles noted, "The greatest problem for mathematicians now is probably the Riemann Hypothesis. But it's not a problem that can be simply stated."  This one seems to be the most accessible introduction we can found out there.

Riemann conjectured that all nontrivial zeros are at Re(z)=1/2. Although this has been shown to be true for more than the first billion nontrivial zeros, the conjecture remains open. A proof would establish new results in number theory, for example on the distribution of primes. The fact that Riemann Hypothesis holds for billions of nontrivial zeros does not guarantee anything. As noted by I. Good and R. Churchhouse in 1968, in the theories of zeta function and of primes distribution, one frequently meets terms like log log x, a function which increases extremely slow. The first nontrivial root not on Re(1/2) might have an imaginary part y such that log log y is of the order say 10. Then y would be 1010,000, a definitely unreachable number, computationally.

Which, to be honest, doesn't mean shit to us but has apparently been catnip to generations of mathematicians who struggle to prove it to this day.

(Personally, we think the best mathematical proof we've ever come across comes from Thomasina Coverly in Tom Stoppard's Arcadia.  She's been given a copy of Fermat's Last Theorem by her tutor Septimus to keep her quiet for a few hours.  As you may know, Fermat was famous for having written in the margin of his copy of Arithmetica "To write the cube of a number as a sum of two cubes, or the fourth power as a sum of two fourth powers, or any power above 2 as a sum of two like powers, is impossible. I have a truly wonderful proof of this fact, but the margin is too narrow to contain it."  Thomasina's reply, prescient given the events of the play, follows:

THOMASINA: Oh!  I see now!  The answer is perfectly obvious.

SEPTIMUS:  This time you may have overreached yourself.

... THOMASINA: There is no proof, Septimus.  The thing that is perfectly obvious is that the note in the margin was a joke to make you all mad.)

Join us tomorrow when we'll be offering up a signed copy of The Indian Clerk for our Friday giveaway.

Posted at 12:01 AM in Indian Clerk Week |

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Comments

j.

Yes. It is unfortunate how hard it is to explain RH. I'd be great if it was like the one you mentioned by Fermat.

Anyways, the idea is that you have a function, "the zeta function" (put some echo here), defined on the complex plane (this is, the function takes complex numbers and gives you complex numbers). This "zeta function" is an "extension" of a real infinite series (one of those infinite sums Ramanujan loved) but I guess we can skip that detail by now.

The point is that if you do z(-2) or z(-4) or z(-6) (and so on, any negative even number), then you get 0. That's more or less easy to see once you know the formula for the zeta function. When they say "trivial zeros", they mean these guys ("the negative even numbers"). However, it was clear the zero function had more zeros (the so-called non-trivial ones) and some properties were known about them.

Parenthesis: To explain these properties, it is probably good to remember a little bit about the complex numbers. A complex number is something of the form a+ib where a and b are real numbers and i is that "unexistent" square root of -1 some of us meet in highschool some in college. We can represent the complex numbers in the cartesian plane assigning a+ib to the pair (a,b). Given a complex number a+ib, the Real Part (Re) of it is a and the Imaginary Part (Im) is b.

Ok, back to RH, the known properties of the non-trivial zeroes of the zeta function were two:

1. The non-trivial zeroes were all located in the vertical strip with Real Part between 0 and 1. This is, comples numbers a+ib where 0 2. If we had a non-trivial zero and we look for the symmetrical complex number (in the cartesian plane) with respect to the line Re(x)=1/2 (The vertical line of complex numbers whose real part is 1/2), then we would find ANOTHER non-trivial zero.

These people studing the zeta function, however, noticed something else: Most of the non-trivial zeros they knew were actually ON the line Re(x)=1/2.

Riemann Hypothesis claims that ALL the non trivial zeros are on the Re(x)=1/2 line. That's it.

The connection between the RH and the distribution of primes is another story and it should probably be told some other time. (Actually, that's the part that's really hard to explain in elementary terms.)

Posted by: j. | August 23, 2007 at 02:49 AM

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