Duality in Analytic Number Theory
eBook - PDF

Duality in Analytic Number Theory

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  3. Available on iOS & Android
eBook - PDF

Duality in Analytic Number Theory

Book details
Table of contents
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About This Book

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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Yes, you can access Duality in Analytic Number Theory by Peter D. T. A. Elliott in PDF and/or ePUB format, as well as other popular books in Mathematics & Number Theory. We have over one million books available in our catalogue for you to explore.

Information

Publisher
Cambridge University Press
Year
1997
ISBN
9780511898778
Topic
Mathematics
Subtopic
Number Theory
Index
Mathematics

Table of contents

  1. Cover
  2. Frontmatter
  3. Contents
  4. Acknowledgements
  5. Preface
  6. Notation
  7. Introduction
  8. Duality and Fourier analysis
  9. Background philosophy
  10. Operator norm inequalities
  11. Dual norm inequalities
  12. Exercises: Including the Large Sieve
  13. The method of the stable dual (1): Deriving the approximate functional equations
  14. The method of the stable dual (2): Solving the approximate functional equations
  15. Exercises: Almost linear, Almost exponential
  16. Additive functions of class ℒα. A first application of the method
  17. Multiplicative functions of the class ℒα: First Approach
  18. Multiplicative functions of the class ℒα: Second Approach
  19. Multiplicative functions of the class ℒα: Third Approach
  20. Exercises: Why the form?
  21. Theorems of Wirsing and HalĂĄsz
  22. Again Wirsing's Theorem
  23. Exercises: The prime number theorem
  24. Finitely distributed additive functions
  25. Multiplicative functions of the class ℒα. Mean value zero
  26. Exercises: Including logarithmic weights
  27. Encounters with Ramanujan's function τ(n)
  28. The operator T on L 2
  29. The operator T on L α and other spaces
  30. Exercises: The operator D and differentiation. The operator T and the convergence of measures
  31. Pause: Towards the discrete derivative
  32. Exercises: Multiplicative functions on arithmetic progressions. Wiener phenomenon
  33. Fractional power Large Sieves. Operators involving primes
  34. Exercises: Probability seen from number theory
  35. Additive functions on arithmetic progressions: Small moduli
  36. Additive functions on arithmetic progressions: Large moduli
  37. Exercises: Maximal inequalities
  38. Shift operators and orthogonal duals
  39. Differences of additive functions. Local inequalities
  40. Linear forms in shifted additive functions
  41. Exercises: Stability. Correlations of multiplicative functions
  42. Further readings
  43. RĂŒckblick (after the manner of Johannes Brahms)
  44. References
  45. Author index
  46. Subject index