Distribution theorems of L-functions. (English) Zbl 0609.10032

Joyner, David

Pitman Research Notes in Mathematics Series, 142. Harlow, Essex, England: Longman Scientific & Technical. Copubl. in the United States with John Wiley & Sons, Inc., New York. VII, 247 p. £16.50 (1986).

In spite of a pretentious introduction the contents of this booklet are as follows. Chapter one gives a short overview of known results on the distribution of zeros of L-functions of different types: Dirichlet L- functions, Artin L-functions, Dirichlet series attached to an automorphic form on the half plane. Also a definition of Langlands’ L-functions for GL(2) is given (just the definition, no philosophy, no context).
Chapters 2 and 3 give, in more detail, distribution theorems and conjectures for the Riemann zeta-function. Then, in chapters 4-6, old work by Selberg is paraphrased and additional results are given. Here the arguments should be valid not only for the Riemann zeta-function (as in Selberg’s work) but also for functions L(s) which satisfy certain axioms, which are sloppily formulated (p. 101-102).
In chapter 7 a known example of an automorphic \(\ell\)-adic motive is discussed with many details on the preliminaries and few on the subject itself.

Reviewer: J.G.M.Mars

MSC:

11Mxx	Zeta and \(L\)-functions: analytic theory
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R42	Zeta functions and \(L\)-functions of number fields
11-02	Research exposition (monographs, survey articles) pertaining to number theory
14A20	Generalizations (algebraic spaces, stacks)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

L-functions; Dirichlet series; automorphic form; distribution theorems; Riemann zeta-function; automorphic \(\ell \)-adic motive