Prime numbers have fascinated mathematicians since there were mathematicians to be fascinated, and the Prime Number Theorem is one of the crowning achievements of the nineteenth century. The theorem answers, in a precise form, a seemingly basic question: how many prime numbers are there?

Up to small thresholds, we may search exhaustively. Up to a hundred, there are 25 primes; up to a thousand, there are 168; up to a million, there are 78,498. The proportion of numbers that are prime seems to be decreasing – from 0.25, to 0.17, to 0.08 – but how quickly? In this podcast, Simon Myerson, Sofia Lindqvist, Jamie Beacom and host Aled Walker reveal the answer, and discuss the collection of mathematical ideas which combine to give the theorem’s first remarkable proof. Listeners who enjoyed Marcus du Sautoy’s ‘The Music of the Primes’ will find similar themes examined in greater detail, but those without any background will find all the necessary terminology developed from first principles.

The story begins with Euclid’s proof of the existence of infinitely many primes. Although this is an argument of infamous elegance, the quantitative aspects are embarrassingly poor. Indeed, the argument only shows that there are at least log log x prime numbers up to a threshold x, and in particular only 5 primes less than a million! In the middle of the nineteenth century, Chebyshev invented methods for detecting many more primes, but he still fell short of the conjectured level of precision. It would take a revolutionary insight of Riemann (pictured), connecting the discrete theory of primes to the continuous theory of mathematical analysis, to uncover the exact distribution of the primes, and to prove the Prime Number Theorem.

This podcast is part of the Secrets of Mathematics series where the pleasure (and occassional) pain of the subject is communicated to a wide audience.

The podcast also forms part of the In Our Spare Time series, in which Oxford Mathematician Aled Walker chairs discussions between various panels of DPhil students, drawn from all the different academic spheres of the university. Current topics range from Oscar Wilde to Dark Matter to Cicero to Medieval Song.