Duality in Analytic Number Theory

Couverture
Cambridge University Press, 13 févr. 1997
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.
 

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Table des matières

Duality and Fourier analysis
1
Background philosophy
16
Operator norm inequalities
18
Dual norm inequalities
25
Including the Large Sieve
32
Deriving the approximate functional equations
48
Solving the approximate functional equations
52
Almost linear Almost exponential
68
The operator T on L 2
159
The operator T on L α and other spaces
169
The operator D and differentiation The operator T and the convergence of measures
183
Towards the discrete derivative
190
Multiplicative functions on arithmetic progressions Wiener phenomenon
205
Fractional power Large Sieves Operators involving primes
211
Probability seen from number theory
232
Small moduli
235

Additive functions of class ℒα A first application of the method
79
First Approach
84
Second Approach
93
Third Approach
101
Why the form?
111
Theorems of Wirsing and Halász
115
Again Wirsings Theorem
122
The prime number theorem
127
Finitely distributed additive functions
133
Multiplicative functions of the class ℒα Mean value zero
139
Including logarithmic weights
148
Encounters with Ramanujans function τn
151
Large moduli
239
Maximal inequalities
254
Shift operators and orthogonal duals
271
Differences of additive functions Local inequalities
275
Linear forms in shifted additive functions
285
Stability Correlations of multiplicative functions
295
Further readings
302
Rückblick after the manner of Johannes Brahms
320
References
321
Author index
333
Subject index
335
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