Fourier Analysis in Number Fields and Hecke's Zeta-functions
Title | Fourier Analysis in Number Fields and Hecke's Zeta-functions PDF eBook |
Author | John Torrence Tate |
Publisher | |
Pages | 55 |
Release | 1950 |
Genre | Algebraic fields |
ISBN |
Fourier Analysis in Number Fields and Hecke's Zeta-function
Title | Fourier Analysis in Number Fields and Hecke's Zeta-function PDF eBook |
Author | John Torrence Tate |
Publisher | |
Pages | |
Release | 1950 |
Genre | Algebraic fields |
ISBN |
Fourier Analysis on Number Fields
Title | Fourier Analysis on Number Fields PDF eBook |
Author | Dinakar Ramakrishnan |
Publisher | Springer Science & Business Media |
Pages | 372 |
Release | 2013-04-17 |
Genre | Mathematics |
ISBN | 1475730853 |
A modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasising harmonic analysis on topological groups. The main goal is to cover John Tates visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries -- technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist. While most of the existing treatments of Tates thesis are somewhat terse and less than complete, the intent here is to be more leisurely, more comprehensive, and more comprehensible. While the choice of objects and methods is naturally guided by specific mathematical goals, the approach is by no means narrow. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups. The text addresses students who have taken a year of graduate-level course in algebra, analysis, and topology. Moreover, the work will act as a good reference for working mathematicians interested in any of these fields.
Fourier Analysis and Zeta Functions on Local Fields
Title | Fourier Analysis and Zeta Functions on Local Fields PDF eBook |
Author | Catriona Glenton |
Publisher | |
Pages | 192 |
Release | 1978 |
Genre | Functions, Zeta |
ISBN |
Spectral Theory of the Riemann Zeta-Function
Title | Spectral Theory of the Riemann Zeta-Function PDF eBook |
Author | Yoichi Motohashi |
Publisher | Cambridge University Press |
Pages | 240 |
Release | 1997-09-11 |
Genre | Mathematics |
ISBN | 1316582507 |
The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.
Harmonic Analysis on Symmetric Spaces and Applications I
Title | Harmonic Analysis on Symmetric Spaces and Applications I PDF eBook |
Author | Audrey Terras |
Publisher | Springer Science & Business Media |
Pages | 353 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461251281 |
Since its beginnings with Fourier (and as far back as the Babylonian astron omers), harmonic analysis has been developed with the goal of unraveling the mysteries of the physical world of quasars, brain tumors, and so forth, as well as the mysteries of the nonphysical, but no less concrete, world of prime numbers, diophantine equations, and zeta functions. Quoting Courant and Hilbert, in the preface to the first German edition of Methods of Mathematical Physics: "Recent trends and fashions have, however, weakened the connection between mathematics and physics. " Such trends are still in evidence, harmful though they may be. My main motivation in writing these notes has been a desire to counteract this tendency towards specialization and describe appli cations of harmonic analysis in such diverse areas as number theory (which happens to be my specialty), statistics, medicine, geophysics, and quantum physics. I remember being quite surprised to learn that the subject is useful. My graduate eduation was that of the 1960s. The standard mathematics graduate course proceeded from Definition 1. 1. 1 to Corollary 14. 5. 59, with no room in between for applications, motivation, history, or references to related work. My aim has been to write a set of notes for a very different sort of course.
Advanced Analytic Number Theory: L-Functions
Title | Advanced Analytic Number Theory: L-Functions PDF eBook |
Author | Carlos J. Moreno |
Publisher | American Mathematical Soc. |
Pages | 313 |
Release | 2005 |
Genre | Mathematics |
ISBN | 0821842668 |
Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. This book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given. The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.