Dirichlet Series

Principles and Methods

Author: S. Mandelbrojt

Publisher: Springer Science & Business Media

ISBN: 9401031347

Category: Mathematics

Page: 176

View: 4481

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.

Encyclopaedia of Mathematics

Coproduct — Hausdorff — Young Inequalities

Author: M. Hazewinkel

Publisher: Springer

ISBN: 1489937951

Category: Mathematics

Page: 963

View: 3409

Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory

Proceedings of the Bretton Woods Workshop on Multiple Dirichlet Series, Bretton Woods, New Hampshire, July 11-14, 2005

Author: Bretton Woods Workshop on Multiple Dirichlet Series

Publisher: American Mathematical Soc.

ISBN: 0821839632

Category: Mathematics

Page: 303

View: 2943

Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series.Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet series, and on developments in the allied fields of automorphic forms and analytic number theory.

Library of Congress Catalog

Books-Subjects, 1970-1974, Set

Author: United States

Publisher: Rowman & Littlefield Publishers

ISBN: 9780874717853

Category: Subject catalogs

Page: N.A

View: 5871

Biographisch-literarisches Handwörterbuch

für Mathematik, Astronomie, Physik mit Geophysik, Chemie, Kristallographie und verwandte Wissensgebiete ...

Author: Johann Christian Poggendorff,Berend Wilhelm Feddersen,Arthur Oettingen,Hans Stobbe,Paul Franz Wilhelm Weinmeister

Publisher: N.A


Category: Science

Page: N.A

View: 8862

Dirichlet's principle

a mathematical comedy of errors and its influence on the development of analysis

Author: A. F. Monna

Publisher: N.A


Category: Mathematics

Page: 138

View: 4003

Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

Author: Dumitru Motreanu,Viorica Venera Motreanu,Nikolaos Papageorgiou

Publisher: Springer Science & Business Media

ISBN: 1461493234

Category: Mathematics

Page: 459

View: 2923

This book focuses on nonlinear boundary value problems and the aspects of nonlinear analysis which are necessary to their study. The authors first give a comprehensive introduction to the many different classical methods from nonlinear analysis, variational principles, and Morse theory. They then provide a rigorous and detailed treatment of the relevant areas of nonlinear analysis with new applications to nonlinear boundary value problems for both ordinary and partial differential equations. Recent results on the existence and multiplicity of critical points for both smooth and nonsmooth functional, developments on the degree theory of monotone type operators, nonlinear maximum and comparison principles for p-Laplacian type operators, and new developments on nonlinear Neumann problems involving non-homogeneous differential operators appear for the first time in book form. The presentation is systematic, and an extensive bibliography and a remarks section at the end of each chapter highlight the text. This work will serve as an invaluable reference for researchers working in nonlinear analysis and partial differential equations as well as a useful tool for all those interested in the topics presented.

Limit Theorems for the Riemann Zeta-Function

Author: Antanas Laurincikas

Publisher: Springer Science & Business Media

ISBN: 9401720916

Category: Mathematics

Page: 306

View: 4956

The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.

Lectures on a Method in the Theory of Exponential Sums

Author: M. Jutila,Tata Institute of Fundamental Research

Publisher: Springer

ISBN: 3540183663

Category: Dirichlet series

Page: 134

View: 1328

These notes are based on the lectures given by the author at the Tata Institute in 1985 on certain classes of exponential sums and their applications in analytic number theory. More specifically, the exponential sums under consideration involve either the divisor function d(n) or Fourier coefficients of cusp forms (e.g. Ramanujan's function #3(n)). However, the "transformation method" presented, relying on general principles such as functional equations, summation formulae and the saddle point method, has a wider scope. Its classical analogue is the familiar "process B" in van der Corput's method, that transforms ordinary exponential sums by Poisson's summation formula and the saddle point method. In the present context, the summation formulae required are of the Voronoi type. These are derived in Chapter I. Chapter II deals with exponential integrals and the saddle point method. The main results of these notes, the general transformation formulae for exponential sums, are then established in Chapter III and some applications are given in Chapter IV. First the transformation of Dirichlet polynomials is worked out in detail, and the rest of the chapter is devoted to estimations of exponential sums and Dirichlet series. The material in Chapters III and IV appears here for the first time in print. The notes are addressed to researchers but are also accessible to graduate students with some basic knowledge of analytic number theory.

Introduction to Partial Differential Equations and Hilbert Space Methods

Author: Karl E. Gustafson

Publisher: Courier Corporation

ISBN: 0486140873

Category: Mathematics

Page: 480

View: 9050

Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.