\(L\)-functions
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.
local distribution of high and low-lying zeros
scientific article; zbMATH DE number 1194158
scientific article