Duality in Analytic Number Theory

Deals with analytic number theory; many new results.

Peter D. T. A. Elliott (Author)

9780521058087, Cambridge University Press

Paperback / softback, published 28 January 2008

360 pages, 80 exercises
22.8 x 15.3 x 2.1 cm, 0.581 kg

' … quite remarkable publication.' European Mathematical Society

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.

Preface
Notation
Introduction
0. Duality and Fourier analysis
1. Background philosophy
2. Operator norm inequalities
3. Dual norm inequalities
4. Exercises: including the large sieve
5. The Method of the Stable Dual (1): deriving the approximate functional equations
6. The Method of the Stable Dual (2): solving the approximate functional equations
7. Exercises: almost linear, almost exponential
8. Additive functions of class La: a first application of the method
9. Multiplicative functions of the class La: first approach
10. Multiplicative functions of the class La: second approach
11. Multiplicative functions of the class La: third approach
12. Exercises: why the form? 13. Theorems of Wirsing and Halász
14. Again Wirsing's theorem
15. Exercises: the Prime Number Theorem
16. Finitely distributed additive functions
17. Multiplicative functions of the class La: mean value zero
18. Exercises: including logarithmic weights
19. Encounters with Ramanujan's function t(n)
20. The operator T on L2
21. The operator T on La and other spaces
22. Exercises: the operator D and differentiation
the operator T and the convergence of measures
23. Pause: towards the discrete derivative
24. Exercises: multiplicative functions on arithmetic progressions
Wiener phenomenon
25. Fractional power large sieves
operators involving primes
26. Exercises: probability seen from number theory
27. Additive functions on arithmetic progressions: small moduli
28. Additive functions on arithmetic progressions: large moduli
29. Exercises: maximal inequalities
30. Shifted operators and orthogonal duals
31. Differences of additive functions
local inequalities
32. Linear forms of additive functions in La
33. Exercises: stability
correlations of multiplicative functions
34. Further readings
35. Rückblick (after the manner of Johannes Brahms)
References
Author index
Subject index.

Subject Areas: Number theory [PBH]