This book contains three texts surveying properties of Riemann's zeta function from different viewpoints. J.-B. Bost explains a proof of the prime number theorem based on the theory of the Fourier transform for distributions. P. Colmez offers a panorama of arithmetic properties of the zeta function (and more general Dirichlet series), ranging from polylogarithms, transcendence results and polyzetas to modular forms, p-adic measures and the p-adic zeta function. Ph. Biane sketches the heuristic relationship between the distribution of zeroes of ζ(1/2+it), the statistical properties of eigenvalues of random unitary matrices , as well as a link between the zeta function and random walks and Brownian processes.

Reviewer:

jnek