An introduction to Artin \(L\)-functions (Q2765412)

This paper is a survey on Artin/Dedekind/Riemann/Dirichlet/Hecke zeta- or \(L\)-functions; it also provides directions in which modern research might go and has gone.NEWLINENEWLINE In Chapter 1, we find motivating examples, in a consequent order of statements. The Riemann zeta-function is mentioned firstly with properties on, respectively, convergence, Euler-product; analytic continuation and functional equation; it ends with the Riemann hypothesis and the theorem of de la Vallée Poussin/Hadamard. Secondly the Dedekind zeta-function is mentioned with the same kind of properties like convergence etc., it also ends with the generalized Riemann hypothesis and the theorem of de la Vallée Poussin/Hadamard. Thirdly the same strategy is given for the classical Dirichlet \(L\)-functions; it ends with Dirichlet's density properties. Fourthly Hecke \(L\)-functions are considered in the same function as before; it ends with results of Hadamard/de la Vallée Poussin/Landau and the prime ideal theorem.NEWLINENEWLINE In Chapter 2 Artin \(L\)-functions are defined and investigated throughout; the so-called completed Artin \(L\)-functions are there, together with the functional equations, induction and restriction properties, the Artin conductor, the Artin root number, adèles and idéles, class fields, the Artin reciprocity law. We find there the Dedekind conjecture on the holomorphy of quotients of Dedekind zeta functions, and results in the literature regarding it. The so-called (strong) Artin conjecture (or Langlands conjecture) is formulated, together with results of Artin-Langlands-Tunnell on 2-dimensional complex matrix groups.NEWLINENEWLINE Now we are in the midth of the paper. The first part gave a rather complete overview what we are dealing with, the second remaining part provides four programs that are running (in 2001), namely the Langlands program, Serre's conjecture, the Brauer-Heilbronn-Stark approach, the Selberg program.NEWLINENEWLINE In Chapter 3 the paper deals with the Brauer-Heilbronn-Stark approach, such as Brauer's famous induction theorem of characters of finite groups, Heilbronn's character, an inequality involving the orders of Artin \(L\)-functions and the order of the Dedekind zeta-function and its consequences.NEWLINENEWLINE In Chapter 4 Selberg's program is considered; we omit technical details here, but we state that the prospects of results are very promising.NEWLINENEWLINE The paper reflects a lecture series given at the Tata Institute of Fundamental Research in 1998. This reviewer found it nice reading.