The classical theory of zeta and \(L\)-functions.
type: Item
instance of: scholarly article
title: The classical theory of zeta and \(L\)-functions.
zbMATH Open document ID: 1275.11117
published in: Milan Journal of Mathematics
publication date: 2011-02-19T00:00:00Z
review text: 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.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11M38
Mathematics Subject Classification ID: 11M50
zbMATH DE Number: 5853581
zbMATH Keywords: zeta function
zbMATH Keywords: Dirichlet L-series
zbMATH Keywords: zeta values
zbMATH Keywords: prime number theorem
zbMATH Keywords: Weil conjectures
MaRDI profile type: Publication
full work available at URL: s00032-010-0121-8
OpenAlex ID: W1969630244
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DOI: 10.1007/S00032-010-0121-8
author: Enrico Bombieri
description: scientific article; zbMATH DE number 5853581