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Marcus du Sautoy
The music of the primes
To mark the start of the new millennium, the Clay Mathematics institute offered $1000000 for the solution of each of seven classic mathematical problems. The longest standing of these problems is Riemann hypothesis, concerning the zeros of the Riemann zeta function ζ(s), which is closely connected to the distribution of the primes. In 'The music of the primes' du Sautoy charts the history of attempts to prove this hypothesis. This is done via biographical details of those involved, so it doesn't require any prior mathematical knowledge. However, one can tell that it is a professional mathematician writing  he clearly knows the subject he is dealing with inside out.
The book starts before the time of Riemann, with Gauss's estimates of how the primes thin out as we go through the integers. These estimates resulted in the prime number conjecture, which Riemann was trying to prove when he invented his zeta function. This conjecture eventually became a theorem, but Riemann's own conjecture remains unproven. Du Sautoy leads us through the attempts of many mathematicans to prove the Riemann hypothesis, showing how often it led to interesting new mathematics, but sometimes to despair for those making the attempt. Indeed it almost reads like a novel, and at times I found it hard to put down  I wanted to see what the next attempt at a proof would lead to.