The classical theory of zeta and \(L\)-functions. (Q627051)

The aim of this article is presenting the basic facts about zeta functions. It begins with Euler's evaluation of \(\zeta(2)\), and mentions, among others, Apéry's proof of the irrationality of \(\zeta(3)\), the functional equation of the Riemann zeta function and Dirichlet's \(L\)-series, Dirichlet's theorem on primes in arithmetic progression and the nonvanishing of \(L(1,\chi)\) for nonprincipal Dirichlet characters, Riemann's theorems and the prime number theorem, the Riemann conjecture and progress in the investigation of zeros on he critical line, generalizations of the zeta function by Dedekind, Hecke, and Artin, Brauer's theorem, connections with modular forms, the Weil conjectures, the motivic look at \(L\)-series, Weil's explicit formulas, and the role of correspondences in the proof of the Weil conjectures. The many historical remarks make reading this article a pleasure even for those who already know these things.