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Duality in Analytic Number Theory

£50.00

Part of Cambridge Tracts in Mathematics

  • Date Published: May 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521058087

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  • 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.

    • Motivates and studies the form as well as proving results
    • Links history into the mathematical narrative
    • Much new material
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    Reviews & endorsements

    ' … a fruitful atempt in finding a general method in Analytic Number Theory.' Monatshefte für Mathematik

    ' … quite remarkable publication.' European Mathematical Society

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    Product details

    • Date Published: May 2008
    • format: Paperback
    • isbn: 9780521058087
    • length: 360 pages
    • dimensions: 229 x 152 x 20 mm
    • weight: 0.53kg
    • contains: 80 exercises
    • availability: Available
  • Table of Contents

    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.

  • Author

    Peter D. T. A. Elliott, University of Colorado, Boulder

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