The ultimate answer

Fans of The Hitchhiker’s Guide to the Galaxy have built up an entire mythology about the number 42, despite Douglas Adams’ (DNA) insistence that the whole thing was just a big joke. Maybe a joke in base-13, but still a joke. The number has in some ways become a self-fulfilling prophecy, in that it pops up everywhere so often that one can’t help but wonder if it isn’t just fan energy driving it.

Well, as the saying goes, truth is stranger than fiction. From Seed Magazine, an article that makes a link between the prime numbers, 42, and the moments of the Riemann zeta function. In other words, the number 42 holds the key to possibly unlocking the secret of the prime numbers themselves, and thus answering an ultimate question of the basic structure of our universe (made physically manifest in the physics of atomic energy levels of heavy elements like Erbium).

So, in a very real sense, the number 42 is the answer to an ultimate question. Whether that is the ultimate question or not is left to the working thinkers [1]. But here’s something else to think about: the zeta function is closely related to the Zipf distribution – which governs the fundamental statistics of the World Wide Web. So perhaps blogging about 42 has deeper meaning than we suspect…

From the article:

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 this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.

But what did this amazing pattern mean for the primes? If Riemann’s discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won’t be a vacuum at one corner and a concentration of molecules at the other.

[…]

It was a chance meeting between physicist Freeman Dyson and number theorist Hugh Montgomery in 1972, over tea at Princeton’s Institute for Advanced Study, that revealed a stunning new connection in the story of the primes—one that might finally provide a clue about how to navigate Riemann’s landscape. They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar.

It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.

[…]

There is an important sequence of numbers called “the moments of the Riemann zeta function.” Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker’s Guide to the Galaxy.

It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.

Oh, Douglas! we knew you were wiser than the Great Prophet Zarquon and Deep Thought combined.

[1] Permission to reveal the Answer to the Ultimate Question of Life, The Universe, and Everything in this blog entry was kindly provided the Amalgamated Union of Philosophers, Sages, Luminaries, and Other Professional Thinking Persons.