Mathematics can be a difficult thing, but I'm very aware that if the maths is difficult, explaining it clearly to others is even more so. There's an even further challenge: to not only explain it, but to explain why it should be of interest. Marcus du Sautoy (a Royal Society Research Fellow) attempts just that with *The music of the primes: why an unsolved problem in mathematics matters*, examining the Riemann Hypothesis.

This may seem adventurous considering that the Riemann Hypothesis is still just that- awaiting proof. Yet by tracing its story from Hilbert's 23 problems in 1900 to today (with a detour into ancient greece through to more recent times once the scene is set), the changes in the way that mathematical problems have been thought about and attacked are also revealed: from inspired proofs of neat but isolated results with numbers; to a drive for rigour; through Gödel and the discovery that we may never be able to prove a statement within a given formulation of number theory; into computer-assisted proofs and the latest non-commutative geometry which gives us the language to describe ever finer subtleties of numbers.

Inevitably though, a story of mathematics is a story of people and this does lead to considerable overlap with other texts. Those who have read *The Man who Loved only Numbers* will recognise many anecdotes relating to Erdös whilst Gödel's impact on mathematics is of course key to Gödel, Escher, Bach. Hardy's use of the Riemann hypothesis to protect him from harm at sea is often recounted but is actually relevant to this tale (as it features the work of Hardy, Littlewood and Ramanujan in considerable detail) whilst Gauss' proof of the sum of the first n integers appears to be a legal requirement for any maths book as it once more features here.

Nonetheless, despite this material shared with a large chunk of my bookshelf, if pressed for an explanation as to why I enjoy pure mathematics this is probably the one I'd reach for as suggested reading. The beauty of number theory is that some of its most challenging problems can be understood by anybody: once you have the idea of a prime, it is easy to understand the issue of how many there are or how they group together. Du Sautoy manages to describe the problem and the ways in which it has been approached very well- in addition to the musical connection, he makes use of the idea of a mountain range as akin to a mathematical landscape. Connections to quantum physics are intriguing and mind-boggling; these are balanced by the easier link between primes, encryption and the safety of your bank balance online.

Best of all, as a mathematician the author manages to relay the sense of excitement that searching out beautiful proofs brings- and throws in enough maths to keep students such as myself amused through the histories. One result in particular had me set the book down in amazement for its sheer simplicity and power- that if the Riemann hypothesis is undecidable then it must be true. He also deftly highlights the differences between maths and other sciences- why computer testing zeros to see if they fall on the line will never be a proof yet proofs of the existence of certain numbers are accepted even if no such number has been turned up so far- and those between modern maths and that of the past, such as the fact that an equation that can generate primes would have been a spectacular find in Euler's day yet adds little to today's quest for themes and connections within mathematics as a whole. Indeed, this search for connections was largely inspired by Riemann's work and so it is fitting that it is the hypothesis bearing his name which continues to drive it to this day.

So it doesn't matter that the hypothesis is as yet unproven for its story so far is a fascinating mix of mathematics, computing, physics and perhaps even music that roams from ancient Greece to modern France via imperial Britain and post-war America. For tracing this history alone I'd recommend Du Sautoy's book; but the insights it offers into the mathematical mindset pushes it beyond a who's-who of numbers and into a compelling read. Oh, and if that doesn't inspire you, there's $1million awaiting should you find a proof of the hypothesis....

"The music of the primes: why an unsolved problem in mathematics matters"- Copyright Marcus du Sautoy 2003. Currently available in hardback ISBN 1-84115-579-9