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42 *IS* The answer to Life, the Universe and Zeta
Posted by
Hemos
on Mon Mar 27, 2006 11:04 AM
from the douglas-adams-was-right dept.
from the douglas-adams-was-right dept.
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.'"
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42 (Score:5, Funny)
242723920317613145364418177377134 (Score:4, Insightful)
Re:242723920317613145364418177377134 (Score:5, Funny)
Parent
You mean (Score:5, Funny)
Re:You mean (Score:5, Funny)
I'm as surprised as you are.
Parent
In more detail (Score:5, Informative)
In fact, the question is:
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
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.Parent
Re:In more detail (Score:5, Funny)
Parent
Re:You mean (Score:5, Interesting)
6x9 = 54 (base 10) = 42 (base 13).
Parent
The answer to everything is a Joke (Score:5, Informative)
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.
Re:The answer to everything is a Joke (Score:3, Funny)
Well it was one of the input parameters, wasn't it? Only thing missing was if he'd drawn it from a sack of scrabble letters. Oh wait, you don't know... *nabs another bit of cheese* This Internet thing is great you know, never see who's at the other end. Well, that ape decendant that lives here should be home soon, guess I better go.
Re:The answer to everything is a Joke (Score:4, Funny)
Brother Maynard: [imdb.com] Skip a bit, Brother.
And three begot the ten thousand things.
Parent
TFA (Score:4, Informative)
by Marcus du Sautoy Posted March 27, 2006 12:40 AM
In 1972, the physicist Freeman Dyson wrote an article called "Missed Opportunities." In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell's equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough--only announced by Einstein some 40 years later.
It is striking that Dyson should have written about scientific ships passing in the night. Shortly after he published the piece, he was responsible for an abrupt collision between physics and mathematics that produced one of the most remarkable scientific ideas of the last half century: that quantum physics and prime numbers are inextricably linked.
This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics.
Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers--any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They also represent one of the greatest challenges in mathematics.
As a mathematician, I've dedicated my life to trying to find patterns, structure and logic in the apparent chaos that surrounds me. Yet this science of patterns seems to be built from a set of numbers which have no logic to them at all. The primes look more like a set of lottery ticket numbers than a sequence generated by some simple formula or law.
For 2,000 years the problem of the pattern of the primes--or the lack thereof--has been like a magnet, drawing in perplexed mathematicians. Among them was Bernhard Riemann who, in 1859, the same year Darwin published his theory of evolution, put forward an equally-revolutionary thesis for the origin of the primes. Riemann was the mathematician in Göttingen responsible for creating the geometry that would become the foundation for Einstein's great breakthrough. But it wasn't only relativity that his theory would unlock.
Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros.
Riemann's discovery was as revolutionary as Einstein's realization that E=mc2. Instead of matter turning into energy, Riemann's equation transformed the primes into points at sea-level in the zeta landscape. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren't scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn't believe t
? 42 is not prime (Score:4, Interesting)
Re:? 42 is not prime (Score:3, Informative)
a) "(...) the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes."
b) "There is an important sequence of numbers called "the moments of the Riemann zeta function.""
So, not only does it not, as far as I understand, claim that the zeroes of the zeta function are actually primes, it also doesn't say that the moments are on the hyp
Re:? 42 is not prime (Score:4, Informative)
Parent
Re:? 42 is not prime (Score:5, Informative)
Well the Riemann zeta function [wikipedia.org] 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 [wikipedia.org] 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.
Parent
How clever! (Score:5, Funny)
Man, those mathematicians are really clever at naming stuff. Next thing you know, they're going to call the places where the function outputs ones, "ones". Will it never end?
Re:How clever! (Score:5, Funny)
Upon trying to describe a stack, he stumbled, paused and said: "Why do you computer people use such strange words like "push" and "pop"? Why not call it 'stick it on the end' and 'take it off the end?' It's so needlessly complicated".
Without a beat, he then writes a bunch of greek symbols on the board, epsilon prime-prime-underbar-hat, muttering on about nondeterministic finite automata and pumping lemmas.
Years ago, I learned never to take any computer science classes from anyone who held only degrees in math, but sadly I had no choice that semester.
Parent
Ooh really funny. (Score:5, Funny)
Re:Ooh really funny. (Score:4, Interesting)
On the other side, every slashdotter thinks they have something funny to say.
Parent
It makes sense (Score:3, Funny)
Oops. So much for encryption (Score:5, Interesting)
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The Zeta function (Score:5, Funny)
Obligatory Alan Turing reference (Score:5, Interesting)
The Slashdot Conjecture (Score:5, Funny)