Diophantine Approximation and Dirichlet Series (Second Edition)

This self-contained book is intended to be read with profit by beginners as well as researchers. It is devoted to Diophantine approximation, the analytic theory of Dirichlet series, and some connections between these two domains, which often occur through the Kronecker approximation theorem. Accordingly, the book is divided into seven chapters, the first three of which present tools from commutative harmonic analysis, including a sharp form of the uncertainty principle, ergodic theory and Diophantine approximation to be used in the sequel. A presentation of continued fraction expansions, including the mixing property of the Gauss map, is given.

Chapters four and five present the general theory of Dirichlet series, with classes of examples connected to continued fractions, the famous Bohr point of view, and then the use of random Dirichlet series to produce non-trivial extremal examples, including sharp forms of the Bohnenblust-Hille theorem. Chapter six deals with Hardy-Dirichlet spaces, which are new and useful Banach spaces of analytic functions in a half-plane. Finally, chapter seven presents the Bagchi-Voronin universality theorems, for the zeta function, and r-tuples of L-functions. The proofs, which mix hilbertian geometry, complex and harmonic analysis, and ergodic theory, are a very good illustration of the material studied earlier.

Reviews

"This is a very nice volume on the analytic theory of classical Dirichlet series... It is a beautiful theory, in which classical analysis often meets Diophantine approximation and number theory, and the authors have done an excellent job selecting material to showcase that beauty. The chosen topics range from introductory material to the frontiers of current research. Yet, the book is mostly self-contained and has rather modest prerequisites: a solid first course in number theory and a year of graduate-level analysis should suffice. Thus, the book can be used both for self-directed studies and as a textbook for a graduate-level topics course in analysis or in analytic number theory. To that end, both the self-taught reader and the instructor should find the end-of-chapter exercises helpful."

-- Mathematical Reviews

Contents

1. A review of commutative harmonic analysis
2. Ergodic theory and Kronecker's theorems
3. Diophantine approximation
4. General properties of Dirichlet series
5. Probabilistic methods for Dirichlet series
6. Hardy spaces of Dirichlet series
7. Voronin type theorems.