Diophantine approximation and Dirichlet series (Q5891264)

Dirichlet series plays an important role in the number theory. The book is devoted to Diophantine approximation and the analytic theory of Dirichlet series. The main focus is the connection between these objects via Kronecker approximation.NEWLINENEWLINEThe book consists of seven chapters, and each of them is continued by few exercises related corresponding chapter. The bibliography has 140 records.NEWLINENEWLINEIn Preface, the short content of the book with emphasis to essential points of investigation is presented.NEWLINENEWLINEIn Chapter 1, the authors give a review of commutative harmonic analysis on locally compact abelian groups with especial attention to the Haar measure, dual group, Plancherel and Pontryagin's theorems. The uncertainty principle for the line or a finite group and the connection with Dirichlet series is discussed.NEWLINENEWLINEChapter 2 contains the basics of ergodic theory with special interest on the applications to the Kronecker theorem, to one/multidimensional equidistribution problems and to Salem numbers.NEWLINENEWLINEChapter 3 discusses on connections between the Diophantine approximation and ergodic theory, and gives a classification of real numbers according to their rate of approximation by rational numbers with controlled denominator.NEWLINENEWLINEChapter 4 deals with the basics of general Dirichlet series, with Perron formulas, with computing the three abscissas of simple, uniform, absolute convergence, and with some comments and examples for the holomorphy abscissa.NEWLINENEWLINEIn Chapter 5, some facts from a theory of random Dirichlet polynomials, multidimensional Bernstein inequality and an approach due to Kahane are presented.NEWLINENEWLINEChaper 6 is devoted to the detailed study of new Banach spaces of Dirichlet series as an extension of initial works by \textit{H. Bohr} [J. Reine Angew. Math. 143, 203--211 (1913; JFM 44.0307.01)], and to open new directions of study infinitedimensional Hankel (Helson) operators. A complete presentation of the Bohnenblust-Hille theorem is given.NEWLINENEWLINEChapter 7 gives a complete proof of the universality theorems of Voronin for the zeta-functions and Bagchi for \(L\)-functions. In the proofs, the Hilbertian geometry, complex and harmonic analysis, and ergodic theory are applied.NEWLINENEWLINEThe book is written in a narrative and friendly to reader style. It is recommended for beginners as well as researchers.