\(L\)-functions
type: Item
instance of: scholarly article
title: \(L\)-functions
zbMATH Open document ID: 0930.11065
published in: Documenta Mathematica
publication date: 1998-08-24T00:00:00Z
full work available at URL: https://eudml.org/doc/229138
full work available at URL: Sarnak.MAN.html
review text: This is a very interesting survey of recent developments of \(L\)-functions theory, having as basic theme the role of families of \(L\)-functions in the study of a given \(L\)-function. The author starts with a brief review of the analytic properties of automorphic \(L\)-functions \(L(s,\pi)\) and of the two central analytic problems in the theory, the Riemann hypothesis and the Ramanujan conjecture. Three main topics are then discussed, with emphasis on the role of families of \(L\)-functions, on function field analogues and on connections with the above mentioned problems. The topics are: (1) sub-convexity bounds for \(L({1\over 2}+it,\pi)\) in the \(t,q\) (conductor) and \(\lambda\) (archimedean size) aspects. (2) Local distribution of high and low-lying zeros of \(L\)-functions and their connections with universal and symmetry laws, respectively. (3) Nonvanishing at special points for families of \(L\)-functions. Numerical computations and several applications are also mentioned, and an impressive list of references is given.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1194158
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: automorphic \(L\)-functions
zbMATH Keywords: numerical computations
zbMATH Keywords: nonvanishing at special points
zbMATH Keywords: local distribution of high and low-lying zeros
zbMATH Keywords: survey
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: Ramanujan conjecture
zbMATH Keywords: function field analogues
zbMATH Keywords: sup-convexity bounds
author: Peter Sarnak
reviewed by: A. Perelli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1194158