Texts and Readings in Mathematics

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 eight 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. Exercises provided at the end of each chapter will be very useful for the reader.

Chapter eight is a new chapter in this second edition. It is devoted to the study of composition operators Cφ on the Hardy space ℋ2 and their complete characterization by Gordon and Hedenmalm. Several misprints and errors have been corrected in the second edition.