Duality in Analytic Number Theory

Peter D. T. A. Elliott

doi:10.1017/CBO9780511983405

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Vaughan, R. C. 1998. Hardy's legacy to number theory. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Vol. 65, Issue. 2, p. 238.

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Elliott, P. D. T. A. 2002. Number Theoretic Methods. Vol. 8, Issue. , p. 119.

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Stakėnas, V. 2006. On some inequalities of probabilistic number theory. Lithuanian Mathematical Journal, Vol. 46, Issue. 2, p. 208.

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Zhang, Wen-Bin 2009. Mean-value theorems and extensions of the Elliott–Daboussi theorem on Beurling’s generalized integers (I). Mathematische Zeitschrift, Vol. 261, Issue. 1, p. 201.

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Elliott, P. D. T. A. 2012. A central limit theorem for Ramanujan’s tau function. The Ramanujan Journal, Vol. 29, Issue. 1-3, p. 145.

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Elliott, P. D. T. A. 2015. Central limit theorems for classical cusp forms. The Ramanujan Journal, Vol. 36, Issue. 1-2, p. 81.

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Stakėnas, Vilius 2015. Jonas Kubilius and genesis of probabilistic number theory. Lithuanian Mathematical Journal, Vol. 55, Issue. 1, p. 25.

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Elliott, P.D.T.A. 2015. Jonas Kubilius and Probabilistic Number Theory Some Personal Reflections. Lithuanian Mathematical Journal, Vol. 55, Issue. 1, p. 2.

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Manstavičius, Eugenijus 2016. From Arithmetic to Zeta-Functions. p. 295.

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Matomäki, Kaisa and Radziwiłł, Maksym 2016. Multiplicative functions in short intervals. Annals of Mathematics, Vol. 183, Issue. 3, p. 1015.

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Elliott, Peter D. T. A 2016. From Arithmetic to Zeta-Functions. p. 109.

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Manstavičius, Eugenijus and Stepas, Vytautas 2017. Variance of additive functions defined on random assemblies. Lithuanian Mathematical Journal, Vol. 57, Issue. 2, p. 222.

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Elliott, P. D. T. A. 2017. Analytic Number Theory, Modular Forms and q-Hypergeometric Series. Vol. 221, Issue. , p. 267.

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Tao, Terence and Teräväinen, Joni 2019. The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures. Duke Mathematical Journal, Vol. 168, Issue. 11,

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Manstavičius, Eugenijus 2020. Sharp bounds for the variance of linear statistics on random permutations. Random Structures & Algorithms, Vol. 57, Issue. 4, p. 1303.

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Mangerel, Alexander P. 2022. Additive functions in short intervals, gaps and a conjecture of Erdős. The Ramanujan Journal, Vol. 59, Issue. 4, p. 1023.

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Sowa, Artur and Fransson, Jonas 2022. Solving the Bose-Hubbard model in new ways. Quantum, Vol. 6, Issue. , p. 728.

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Karbonskis, Arvydas and Manstavičius, Eugenijus 2024. Variance of a strongly additive function defined on random permutations. Lithuanian Mathematical Journal, Vol. 64, Issue. 3, p. 302.

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

Publisher:: Cambridge University Press
Online publication date:: November 2011
Print publication year:: 1997
Online ISBN:: 9780511983405
DOI:: https://doi.org/10.1017/CBO9780511983405

Subjects:: Mathematics (general), Mathematics, Number Theory, Discrete Mathematics Information Theory and Coding
Series:: Cambridge Tracts in Mathematics (122)

51.99 (GBP)

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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.

‘ … a fruitful atempt in finding a general method in Analytic Number Theory.’

Source: Monatshefte für Mathematik

‘ … quite remarkable publication.’

Source: European Mathematical Society

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Duality in Analytic Number Theory

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Acknowledgements
pp vii-viii

Preface
pp ix-ix

Notation
pp x-xii

Introduction
pp xiii-xviii

0 - Duality and Fourier analysis
pp 1-15

1 - Background philosophy
pp 16-17

2 - Operator norm inequalities
pp 18-24

3 - Dual norm inequalities
pp 25-31

4 - Exercises: Including the Large Sieve
pp 32-47

5 - The method of the stable dual (1): Deriving the approximate functional equations
pp 48-51

6 - The method of the stable dual (2): Solving the approximate functional equations
pp 52-67

7 - Exercises: Almost linear, Almost exponential
pp 68-78

8 - Additive functions of class ℒα. A first application of the method
pp 79-83

9 - Multiplicative functions of the class ℒα: First Approach
pp 84-92

10 - Multiplicative functions of the class ℒα: Second Approach
pp 93-100

11 - Multiplicative functions of the class ℒα: Third Approach
pp 101-110

12 - Exercises: Why the form?
pp 111-114

13 - Theorems of Wirsing and Halász
pp 115-121

14 - Again Wirsing's Theorem
pp 122-126

15 - Exercises: The prime number theorem
pp 127-132

16 - Finitely distributed additive functions
pp 133-138

17 - Multiplicative functions of the class ℒα. Mean value zero
pp 139-147

18 - Exercises: Including logarithmic weights
pp 148-150

19 - Encounters with Ramanujan's function τ(n)
pp 151-158

20 - The operator T on L2
pp 159-168

21 - The operator T on Lα and other spaces
pp 169-182

22 - Exercises: The operator D and differentiation. The operator T and the convergence of measures
pp 183-189

23 - Pause: Towards the discrete derivative
pp 190-204

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Duality in Analytic Number Theory

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