music of the primes (1)

This semester, I’m running a 3rd year course on Marcus du Sautoy’s The music of the primes. The concept being that students may suggest topics, merely sketched in the book, and then we’ll go a little deeper into them.

I’ve been rather critical about the book before, but, rereading it last week (and knowing a bit better the limitations of bringing mathematics to the masses…) I think du Sautoy did a great job. Sure, it focusses too much on people and places and too little on mathematics, but that goes with the format.

I wanted to start off gently by playing the open-university dvd-series so that students would have a very rough outline of the book from the very start (as well as a mental image to some of the places mentioned, such as Bletchley Park, the IAS, Gottingen…). However, the vagueness of it all seemed to work on their nerves … in particular the trumpet scenes

Afterwards, they demanded that I should explain next week what on earth the zeroes of the Riemann zeta function had to do with counting primes and what all this nonsensical ‘music of the primes’ was about.

Well, here is the genuine music of the primes (taken from the Riemann page by Jeffrey Stopple whose excellent introductory text A Primer of Analytic Number Theory I’ll use to show them some concrete stuff (they have their first course on complex analysis also this semester, so I cannot go too deep into it).




Jeffrey writes “This sound is best listened to with headphones or external speakers. For maximum effect, play it LOUD.” But, what is the story behind it?

The Von Mangoldt function $\Lambda(n) $ assigns $log(p) $ whenever $n=p^k $ is a prime power and zero otherwise. One can then consider the function

$\Psi(x) = \frac{1}{2}(\sum_{n < x} \Lambda(n) + \sum_{n \leq x} \Lambda(n)) $

which makes a jump at prime power values and the jump-size depends on the prime. Here is a graph of its small values

It’s not quite the function $\pi(x) $ (counting the number of primes smaller than x) but it sure contains enough information to obtain this provided we have a way of describing $\Psi(x) $.

The Riemann zeta function (or rather $~(s-1)\zeta(s) $) has two product descriptions, the Hadamard product formula (running over all zeroes, both the trivial ones at $-2n $ and those in the critical strip), which is valid for all complex s and the Euler product valid for all $Re(s) > 1 $. This will allow us to calculate in two different ways $\zeta'(s)/\zeta(s) $ which in turn allows us to have an explicit description of $\Psi(s) $ known as the Von Mangoldt formula

$\Psi(x) = x – \frac{1}{2}log(1 – \frac{1}{x^2}) – log(2 \pi) – \sum_{\rho} \frac{x^{\rho}}{\rho} $

where only the last term depends on the zeta-zeroes $\rho $ lying in the critical strip (and conjecturally all lying on the line $Re(x) = \frac{1}{2} $. The first few terms (those independent of the zeroes) give a continuous approximation of $\Psi(x) $ but how on earth can we get from that approxamation (on the left) to the step-like function itself (on the right)?



We can group together zeta-zeroes $\rho=\beta + i \gamma $ with their comlex conjugate zeroes $\overline{\rho} $ and then one shows that the attribution to the Von Mangoldt formula is

$\frac{x^{\rho}}{\rho} + \frac{x^{\overline{\rho}}}{\overline{rho}} = \frac{2 x^{\beta}}{| \rho | }cos(\gamma log(x) – arctan(\gamma/\beta)) $

Ignoring the term $x^{\beta} $ this is a peridodic function with amplitude $2/| \rho | $ (so getting smaller for larger and larger zeroes) and period $2\pi/ \gamma $. If the Riemann hypothesis holds (meaning that $\beta=1/2 $ for all zeroes) one can even split a term in this contribution of every zero as a sort of ‘universal amplitude’. What is left is then a sum of purely periodic functions which a physicist will view as a superposition of (sound) waves and that is the music played by the primes!

Below, a video of the influence of adding the first 100 zeroes to a better and better approximation of $\Psi(x) $ (again taken from the Riemann page by Jeffrey Stopple). Surely watching the video will convince anyone of the importance of the Riemann zeta-zeroes to the prime-counting problem..




Blackle

There seems to be a slight chance that the next US-administration may (finally) be joining the rest of the civilized world and sign the Kyoto-treaty. Here’s an appeal to Flock and other webbrowsers : please add blackle.com to our Search Engine Preferences!

The idea is simple : you Google as you’d do anyway but … you save a lot of energy. Via PD2 (for Pseudonymous Daughter 2).

Archimedes’ stomachion

The Archimedes codex is a good read, especially when you are (like me) a failed archeologist. The palimpsest (Greek for ‘scraped again’) is the worlds first Kyoto-approved ‘sustainable writing’. Isn’t it great to realize that one of the few surviving texts by Archimedes only made it because some monks recycled an old medieval parchment by scraping off most of the text, cutting the pages in half, rebinding them and writing a song-book on them…

The Archimedes-text is barely visible as vertical lines running through the song-lyrics. There is a great website telling the story in all its detail.

Contrary to what the books claims I don’t think we will have to rewrite maths history. Didn’t we already know that the Greek were able to compute areas and volumes by approximating them with polygons resp. polytopes? Of course one might view this as a precursor to integral calculus… And then the claim that Archimedes invented ordinal calculus. Sure the Greek knew that there were ‘as many’ even integers than integers… No, for me the major surprise was their theory about the genesis of mathematical notation.

The Greek were pure ASCII mathematicians : they wrote their proofs out in full text. Now, here’s an interesting theory how symbols got into maths… pure laziness of the medieval monks transcribing the old works! Copying a text was a dull undertaking so instead of repeating ‘has the same ratio as’ for the 1001th time, these monks introduced abbreviations like $\Sigma $ instead… and from then on things got slightly out of hand.

Another great chapter is on the stomachion, perhaps the oldest mathematical puzzle. Just a few pages made in into the palimpsest so we do not really know what (if anything) Archimedes had to say about it, but the conjecture is that he was after the number of different ways one could make a square with the following 14 pieces

People used computers to show that the total number is $17152=2^8 \times 67 $. The 2-power is hardly surprising in view of symmetries of the square (giving $8 $) and the fact that one can flip one of the two vertical or diagonal parts in the alternative description of the square

but I sure would like to know where the factor 67 is coming from… The MAA and UCSD have some good pages related to the stomachion puzzle. Finally, the book also views the problema bovinum as an authentic Archimedes, so maybe I should stick to my promise to blog about it, after all…

i’ll take rerun requests

If you write a comment-provoking post (such as that one), you’d better deal with the reactions.

As often, the bluntest comment came from the Granada-Antwerp commuter (aka “mewt” for ‘memories of a weird traveler’…)

Javier :

Concerning the participation on the math-related posts, it is true that what you write has become more readable along the years, but yet, being able to read one of your math posts and catch the idea of what is going on (which I think is a great thing to do) is one thing. Actually understanding the details is a completely different one. And possibly most people thinks that commenting around when you only got the general idea (if any) of some math topic would be rather bold.

Personally, with your 2 last posts concerning Connes-Bost systems I am interested on understanding the story in full detail, so I printed your first post, took it home, read it carefully, made all the computations on my own (not that I dont trust yours, but you never know!) and before I had finished getting a sound impression of what was going on, the second part was already online, so had to go through the same process (in top of usual duties) just to keep your rythm. If things go as usual, by the time I am ready to make any sensible comments, you’ll be already bored of the topic and have switched to something else, so it won’t make much sense commenting at all!If its comments what you’re lasting for, write short, one-idea posts, rather than long, technically detailed ones.

The good news is that my posts become slightly more understandable. But all things can be improved… so, here’s a request :

If there is this one post you’d love to understand if only you knew already the material I subconsciously assumed, tell me or leave a comment!

and I’ll try to improve on it…

Oxen of the Sun

The Oxen of the Sun (of the Problema Bovinum) is one of the most difficult chapters in Joyce’s Ulysses. Ulysses is the 1904 version of Homer’s Odyssey so the Oxen appear also in his Book XII :

And thou wilt come to the isle Thrinacia. There in great numbers feed the kine [cattle] of Helios and his goodly flocks, seven herds of kine and as many fair flocks of sheep, and fifty in each.

Homer must have suffered from an acute form of innumeracy as the minimal solution to the cattle problem gives as the total number the smallest integer greater than

$\frac{25194541}{184119152} (109931986732829734979866232821433543901088049+ $

$50549485234315033074477819735540408986340
\sqrt{4729494})^{4658} $

a number whose actual digits take up 47 pages, one of the most useless pieces of mathematical wall-paper!

lieven le bruyn's (old) blog