Duality in Analytic Number Theory

by Peter D. T. A. Elliott

• ISBN13: 9780521058087
• ISBN10: 0521058082
• Edition: Revised
• Format: Paperback
• Copyright: 2008-01-28
• Publisher: Cambridge University Press

Summary

In this stimulating book, aimed at researchers both established and budding, Peter Elliott demonstrates a method and a motivating philosophy that combine to cohere a large part of analytic number theory, including the hitherto nebulous study of arithmetic functions. Besides its application, the book also illustrates a way of thinking mathematically: historical background is woven into the narrative, variant proofs illustrate obstructions, false steps and the development of insight, in a manner reminiscent of Euler. It is shown how to formulate theorems as well as how to construct their proofs. Elementary notions from functional analysis, Fourier analysis, functional equations and stability in mechanics are controlled by a geometric view and synthesized to provide an arithmetical analogue of classical harmonic analysis that is powerful enough to establish arithmetic propositions until now beyond reach. Connections with other branches of analysis are illustrated by over 250 exercises, structured in chains about individual topics.

Table of Contents

Preface

Notation

Introduction

Duality and Fourier analysis

Background philosophy

Operator norm inequalities

Dual norm inequalities

Exercises: including the large sieve

The Method of the Stable Dual (1): deriving the approximate functional equations

The Method of the Stable Dual (2): solving the approximate functional equations

Exercises: almost linear, almost exponential

Additive functions of class La: a first application of the method

Multiplicative functions of the class La: first approach

Multiplicative functions of the class La: second approach

Multiplicative functions of the class La: third approach

Exercises: why the form

Theorems of Wirsing and Halfsz

Again Wirsing's theorem

Exercises: the Prime Number Theorem

Finitely distributed additive functions

Multiplicative functions of the class La: mean value zero

Exercises: including logarithmic weights

Encounters with Ramanujan's function t(n)

The operator T on L2

The operator T on La and other spaces

Exercises: the operator D and differentiation; the operator T and the convergence of measures

Pause: towards the discrete derivative

Exercises: multiplicative functions on arithmetic progressions

Wiener phenomenon

Fractional power large sieves; operators involving primes

Exercises: probability seen from number theory

Additive functions on arithmetic progressions: small moduli

Additive functions on arithmetic progressions: large moduli

Exercises: maximal inequalities

Shifted operators and orthogonal duals

Differences of additive functions; local inequalities

Linear forms of additive functions in La

Exercises: stability; correlations of multiplicative functions

Further readings

R8ckblick (after the manner of Johannes Brahms)

References

Author index

Subject index

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