The Music of the Primes
This afternoon I finished reading a remarkable book called ‘The Music of the Primes’ by Marcus du Sautoy. It was actually a Christmas present from my boyfriend, and a rather sweet one at that, because I told him he didn’t need to buy me anything else, yet he still did :) I am very glad he ignored my protestations and went ahead; otherwise I would have missed out on a real treat.
Having studied mathematics at university, I am always sadly excited when I get the chance to read anything about numbers, but more often than not I find that maths books are disappointing. Either they are so obscurely complicated that no one of only average intelligence who is trying to conduct a normal life could possibly devote the time and energy required to understand them, or the author has decided that his work needs to be accessible to small children and spends a patronising chapter explaining what an integer is. Without using the word integer, obviously, lest that upset people, and probably suggesting the reader imagine they are part of an African tribe and counting out beans. This book, which falls into neither of the above categories, was a refreshing change; an example of that all too rare species, books which deal with complex mathematics yet are pleasurable to read 
Written in a clear and readable style, it includes just enough detail to be accurate and comprehensive, without including so much as to be dull and incomprehensible. Proofs, equations and graphs are included only where strictly necessary and in circumstances where they add to the understandability of the text as opposed to destroying it. It is probably fair to say that if you don’t have a certain level of interest in and thus knowledge of mathematics that you aren’t going to find it a riveting read and may not see it through to the end, but there is just the right mix of mathematical concepts with historical and biographical facts to keep the reader’s attention. Part of my enjoyment undoubtedly came from the fact that this was a subject I had studied at uni and therefore I was able to consult my own notes on Number Theory when I came across results which I knew I had learnt to prove during my degree but could no longer quite remember. This gave me an additional insight into the book, and the book in turn gave me an additional insight into my degree, describing the real figures and events behind the sometimes dull series of lemmas and theorems with which our lectures were filled.
The book tells the story of some of the most amazing numbers in mathematics; the primes. I still remember the first definition I ever learned at university, and my complete and total horror at the fact that something as simple as p not having any factors except itself and one needed such a convoluted array of symbols to express it. What was wrong with words?!
In any case, ‘The Music of the Primes’ is a detailed account of human attempts to understand prime numbers. Some of the most eminent names in mathematics have worked in this field of number theory, from Euclid and his beautiful proof that there are infinitely many prime numbers, to Gauss and Euler, Hardy and Ramanujan. My hero Georg Cantor and his continuum hypothesis (that there are only two sizes of infinity represented by the real numbers) even got the odd mention, and I do in fact remember reading a translation of his work on transfinite primes once, but I don’t seem to have any notes on it. The most significant figure in the entire saga, however, is Bernhard Riemann who proposed the fiendish Riemann hypothesis.
I have to say, anything I have ever come across by Riemann, I have failed to understand. He seems to be one of those men who have a vast amount of mathematics named after him, and I can remember a curious course I took called Further Complex Variable Theory, where his name cropped up with significant frequency. The course was curious because the name implied there should have been a prior course called Complex Variable Theory for this to be further to, only there hadn’t been. It was taught by a very lovely man who unfortunately had no idea how to teach, mumbled terribly, and through an unfortunate accident in a first year tutorial happened to know my name, which meant he used to ask me questions whenever he got desperate. We learnt about Riemann sums and Riemann integrals and once there was some mention of the Riemann sphere, which threw me completely. By the second lecture I realised I wasn’t going to understand a word of it, and my marks in the homework questions were somewhat depressing. Fortunately the lecturer provided printed notes, which I memorised when it came to revision, and via what must have been some pretty spectacular moderation, eventually passed with 88% 
I felt a bit more sympathetic to Riemann after reading the book and have almost forgiven him for the mental contortions his spheres have caused me. Generations of mathematicians have, however, wrestled with attempted proofs of his famous hypothesis and without exception failed. The Riemann hypothesis is one of the most important unsolved problems in mathematics, and has only increased in importance since Riemann proposed it in 1859. The more attempts which have been made to either prove or disprove it, the more has seemed to ride on whether or not it is true. Stated as simply as possible, it is the conjecture that for the Riemann zeta function, all non-trivial zeros will have a real part equal to ½. What the Riemann zeta function is, I’m not sure I feel equal to explaining to you; it’s a function of sum variable x, defined as the sum of an infinite series, and in a real domain it is fairly well behaved, but from what I recall, extending it into the complex domain involves horrible things like cuts in planes and poles and other things which washed over my head in further complex variable theory. The book itself describes it in terms of a landscape in four dimensions and tries to give helpful diagrams, though I confess to not completely understanding those either, because three dimensions are normally sufficient to flummox me. I have always disliked the zeta function in principal anyway, because I have a complete incapability of drawing the Greek letter zeta :cry:
In any case, the extent to which you understand the zeta function will not have much impact on your understanding of the book. In a way I think the story it tells is a profoundly sad one; so many people who have dedicated their lives to trying to prove this hypothesis, some who have even lost their lives, by physically dying or going mad, and yet the book comes to an end and the hypothesis is still unproven. No one has even managed to prove that it can’t be proved, as Gödel and Cohen managed with the Continuum Hypothesis, and no computer has managed to find a counter example, a non-trivial zero which has a real part not equal to a half. It seems somehow depressing 
As the book indicates, a proof of the Riemann hypothesis is not just a random happening in abstract mathematics about which we do not need to care, but something which could affect all of our lives. You probably already know that much encryption (for example credit card numbers) which is carried out during internet transactions, relies on the fact that it is incredibly difficult to break a large number down into two prime factors. If the work which is being done to attempt to prove the Riemann hypothesis ever sheds any light on a simpler way to do this, e-commerce would suffer a significant setback. The chapters on cryptography are pretty cool and not at all difficult to understand. One of the later chapters went slightly over my head when it started talking about quantum physics, but that’s probably my fault for being too stupid to understand physics. Physics upsets me 
Anyway I thoroughly enjoyed it, so much so that I am going to investigate getting a copy of Hardy’s ‘A Mathematician’s Apology’. A friend of mine recommended it to me ages ago, but I’ve not had the motivation to consider it until now
Tags: Books, mathematics, Music of the Primes, primes