42 *IS* The answer to Life, the Universe and Zeta

Venusian Treen writes "In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. The gist is that energy levels in the nucleus of heavy atoms can tell us about the distribution of zeros in Riemann's zeta function - and hence where to find prime numbers. This article discusses this connection, and introduces two physisicts who tell us 'why the answer to life, the universe and the third moment of the Riemann zeta function should be 42.'"

Comments

The answer to everything is a Joke

[digitaldc]

Douglas Adams was asked many times during his career why he chose the number forty-two. Many theories were proposed, but he rejected them all. On November 2, 1993, he gave an answer on alt.fan.douglas-adams:
The answer to this is very simple. It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one. Binary representations, base thirteen, Tibetan monks are all complete nonsense. I sat at my desk, stared into the garden and thought '42 will do' I typed it out. End of story.

Tao Te Ching, Chapter 42:

The Tao begot one. One begot two. Two begot three. And three begot the ten thousand things. The ten thousand things carry yin and embrace yang. They achieve harmony by combining these forces. Men hate to be "orphaned," "widowed," or "worthless," But this is how kings and lords describe themselves. For one gains by losing and loses by gaining. What others teach, I also teach; that is: "A violent man will die a violent death! " This will be the essence of my teaching.

In more detail

[l2718]

In fact, the question is:

What is the arithmetic factor in the asymptotics of the third moment of the Riemann zeta-function?

In more detail: If you integrate the nth power of the absolute value of the Riemann zeta function on the the critical line between heights -T and T and divide by 2T, you will get a sort of nth moment on average. Random matrix theory predicts the growth of this function to be asymptotic to a "geometric factor" (coming from an integral over the unitary group) times the n^2 power of the logarithm of T. It turned out that the random matrix theory prediction is off by an "arithmetic" factor, so that the correct asymptotics is

a(n)g(n) (log T)^(n^2)

where g(n) is the geometric factor from above and a(n) is a rational number. The article is about the prediction a(3)=42.

Re:? 42 is not prime

[Coryoth]

Are there any mathematicians who can explain how a non-prime is the third riemann moment in the string of riemann zeros?

Well the Riemann zeta function is an otherwise innocuous looking function where zeta(z) = 1 + 1/(2^z) + 1/(3^z) + 1/(4^z) + ...

It has some surprising and intriguing properties however. One of the more interesting is that it ends up appearing inside a formula to approximate the prime number counting function (which counts the number of primes less than n). Because of the way it appears in the integral that provides the formula (as log(1/zeta(z))) and "poles" (essentially points where the function shoots of to infinity like asymptotes, except on the complex plane) of the function being integrated are vitally important for determining these particular kinds of integral (complex path integrals) it turns out that determining when the Riemann zeta funtion is zero has a lot to say about the distribution of prime numbers.

This means we've converted the problem from studying the distribution of prime numbers (very hard) to studying the distribution of the zeros of a particular function (hard, but a definite improvement). So what can we say about the distribution of zeros of the Riemann zeta funtion? Well without actually knowing where all the zeros are we can at least potentially talk about the moments of the distribution which is basically just a series of statistical measures. The first moment of a distribution is the mean, the second moment is the variance. What they have found is the third moment, the next step up from the variance, of the distribution of zeros of the Riemann zeta function - whih, as we've seen, in deeply connected to the distribution of prime numbers.

The third moment of ther distribution of zeros of the Riemann zeta function can thus be any number: it isn't required to be prime; it is simply a measure describing properties of the distribution. Exactly what that number is though, can actually say a lot about how prime numbers are distributed.

Jedidiah.