Duality in Analytic Number Theory

Deals with analytic number theory; many new results.

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.

In this stimulating book, Peter Elliott demonstrates a method and a motivating philosophy that combine to cohere a large part of analytic number theory. He also illustrates a way of thinking mathematically and shows how to formulate theorems as well as construct their proofs.