Dirichlet Series: Principles and Methods

It is not our intention to present a treatise on Dirichlet series. This part of harmonic analysis is so vast, so rich in publications and in 'theorems' that it appears to us inconceivable and, to our mind, void of interest to assemble anything but a restricted (but relatively complete) branch of the theory. We have not tried to give an account of the very important results of G. P6lya which link his notion of maximum density to the analytic continuation of the series, nor the researches to which the names of A. Ostrowski and V. Bernstein are intimately attached. The excellent book of the latter, which was published in the Collection Borel more than thirty years ago, gives an account of them with all the clarity one can wish for. Nevertheless, some scattered results proved by these authors have found their place among the relevant results, partly by their statements, partly as a working tool. We have adopted a more personal point of view, in explaining the methods and the principles (as the title of the book indicates) that originate in our research work and provide a collection of results which we develop here; we have also included others, due to present-day authors, which enable us to form a coherent whole.

Table of Contents:
I/Sequences of Exponents and Associated Sequences. Elementary Theorems on the Coefficients and on Convergence.- I.1. Ascending Sequences of Positive Numbers.- I.2. General Properties of Convergence.- I.3. The Calculus of Coefficients and Some Important Combinations of Coefficients.- II/Inequalities Concerning the Coefficients.- II. 1. Inequalities Corresponding to the Arithmetic Character of the Exponents.- II.2. A General Inequality for the Coefficients Corresponding to Finite Upper Density of the Exponents.- III/Theorems of Liouville-Weierstrass-Picard Type of Arithmetical Character and of General Character.- III. 1. Theorems of Arithmetical Type.- III. 2. Liouville-Picard Theorems of General Type.- IV/Singularities of Functions Represented by a Taylor Series.- IV. 1. Singularities of Taylor Series.- V/Composition of Singularities.- V.l. Composition of Hadamard Type and Generalizations.- V.2. Composition of Functions of Slow Growth.- V.3. ‘Fictitious Composition’ of Singularities.- V.4. Composition of Functions of Rapid Growth.- V.5. Composition of ‘Hurwitz Type’.- VI/Some Applications of the Principles for Analytic Continuation.- VI.1. Arithmetic Properties of the Exponents and the Analytic Continuation.- VI.2. Analytic Continuation of General Dirichlet Series.- VI.3. Entire Functions Represented by Dirichlet Series.- VI.4. Some Applications of the Composition Theorems.- VII/On the Behaviour of the Remainders of a Dirichlet Series in the Domain of Existence of the Function. Applications.- VII.1. Evaluation of the Remainders of (A, ?).- VII.2. Applications of the Results of Chapter III and of VI. to Series Meromorphic on an Interval of the Axis of Convergence 124.- VIII/Applications to the Generalized Rie-Mann Functional Equation.- VIII.1. Number of Independent Solutions. Links between Exponents.- IX/Influence of Arithmetical Properties of the Exponents of a Dirichlet Series on its Analytic Continuation.- IX.1. Isolated Fractional Parts of the Exponents and the Possibility of Analytic Continuation.- IX.2. Relations Between Isolation of the Fractional Parts of the Exponents and the Distribution of Singularities.- IX.3. Variation of the Exponents and the Analytic Continuation.- X/Bibliographical Notes.- Note.