Paul Simon's 1983 song "When Numbers Get Serious" gives musical expression to Einstein's wise observation that "not everything that counts can be counted, and not everything that can be counted counts". For the pure mathematicians who study numbers for their own sake, the numbers that count highest in their affection are the primes. A prime is a number that can be divided without remainder only by itself and by one, for example, 2, 3, 5, 7, 11, 13 and 17. Mathematics is a universal language, independent of culture, so if we Earthlings detected extraterrestrial signals showing knowledge of the primes, we could infer the existence of another numerate civilisation.

Even after millennia of intense mathematical work, no one has been able to identify a pattern among these numbers. However, a tantalising insight into the pattern was produced almost 150 years ago by the great German mathematician Bern hard Riemann when he published a remarkable hypothesis that - if true - says that the primes have music in them.

The connection between music and the primes is not trivial, but it is made plausible to the mathematically terrified in this delightfully entertaining book by the Oxford University mathematician Marcus du Sautoy. He begins by describing the three-dimensional landscape that Riemann constructed to give insights into the primes, with each point at sea level corresponding to a musical note. It is the combination of these notes, each at just the right volume, that gives rise to the music of the primes. If Riemann's landscape is not exactly right, the effect of a single wrong point at sea level would be like a sonorous fart issuing from a tuba during the andante of the Jupiter symphony. The music would be destroyed.

Du Sautoy has said that he wanted this, his first popular book, to "read like a novel from the opening page". He has certainly been successful in setting up a compelling dramatis personae of mathematicians, with every character vividly illuminated by anecdotes and felicitous comment. The perfectionist Riemann comes across as an especially intriguing figure. His death at the age of 39 deprived us of one of the 19th century's most original minds. The part of his papers that survived a tragically over-zealous clear-out by his housekeeper continue to yield new mathematical gems.

Sometimes, however, the narrative flags. Du Sautoy is wont to have too much faith in our ability to recall the details of Riemann's hypothesis, and at one point the book almost descends into a series of mathematical vignettes. But he recovers well, with some entertaining tales of how number theory has lost its purity as practical uses for it have been discovered. For example, the theory of prime numbers has led to ways of encrypting credit-card transactions on the Web, keeping our personal business away from prying eyes.

Prime numbers turn up in nature, too. Scientists have often found that the universe dances to the most beautiful mathematical tunes, but it was initially a surprise to find that the spacings of the energy values of some atomic nuclei look pretty much identical to those of the prime numbers. Later the physicist Sir Michael Berry (coiner of the phrase "music of the primes") showed that the same spacing also occurs in quantum billiards, between the energy values of a quantum particle rattling chaotically around in a container the shape of a soccer stadium.

With practical applications of prime number theory so abundant, the theory is clearly no longer the preserve of the pure mathematicians. Scientists and even computer programmers are now making important contributions. However, most mathematicians believe that the Riemann hypothesis will eventually be cracked by someone of their breed, working in the old-fashioned way with a pen, paper and large waste bin.

For whoever is brilliant enough to prove the hypothesis, immortality awaits. Although it is conceivable that our culture could develop in a way such that Mozart's music one day has no value, if there is music in the primes, it is eternal. As Paul Simon says in that song: "Serious numbers will speak to us always."