Dirichlet Series: Principles and MethodsIt 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. |
Contents
SEQUENCES OF EXPONENTS | 1 |
INEQUALITIES CONCERNING THE COEFFICIENTS | 18 |
CHAPTER IIITHEOREMS Of LiouvilleWEIERSTRASSPICARD | 31 |
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abscissa absolute convergence analytic continuation assume axis of convergence b₁ b₂ Chapter circle of convergence coefficients compact set Consequently contains converges absolutely converges uniformly corresponds defined denote Dirichlet series disk domain e-ns entire function exponents Fa(z finite following properties following theorem function f(s function represented given Hadamard's theorem half-plane Hence holomorphic function inequality isolated LD(G Lemma lim log lim sup log log log m₁ modulo plane polynomial positive integers possible singularities proof of Theorem quantity R)-order radius of convergence real number regular point replaced respect Riemann functional satisfies sequence of positive simple poles singular point singular set singularity of F So₂ starred curve statement sufficiently large sufficiently small Suppose t₁ Taylor series Taylor-D tends to infinity tends to zero Theorem IX.1.1 tion uniform upper density values write λη σ₁