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Introduction to Probabilistic Number Theory [Kõva köide]

(Swiss Federal Institute of Technology, Zürich)
  • Formaat: Hardback, 250 pages, kõrgus x laius x paksus: 150x230x25 mm, kaal: 550 g, Worked examples or Exercises
  • Sari: Cambridge Studies in Advanced Mathematics
  • Ilmumisaeg: 06-May-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108840965
  • ISBN-13: 9781108840965
Teised raamatud teemal:
Introduction to Probabilistic Number TheorySuurem pilt
  • Formaat: Hardback, 250 pages, kõrgus x laius x paksus: 150x230x25 mm, kaal: 550 g, Worked examples or Exercises
  • Sari: Cambridge Studies in Advanced Mathematics
  • Ilmumisaeg: 06-May-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108840965
  • ISBN-13: 9781108840965
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781108840965.html
Märksõnad:
"Despite its seemingly deterministic nature, the study of whole numbers, especially prime numbers, has many interactions with probability theory, the theory of random processes and events. This surprising connection was first discovered around 1920, but in recent years, the links have become much deeper and better understood. Aimed at beginning graduate students, this textbook is the first to explain some of the most modern parts of the story. Such topics include the Chebychev bias, universality of the Riemann zeta function, exponential sums, and the bewitching shapes known as Kloosterman paths. Emphasis is given throughout to probabilistic ideas in the arguments, not just the final statements, and the focus is on key examples over technicalities. The book develops probabilistic number theory from scratch, with short appendices summarizing the most important background results from number theory, analysis, and probability, making it a readable and incisive introduction to this beautiful area of mathematics"--

Arvustused

'an excellent resource for someone trying to enter the field of probabilistic number theory' Bookshelf by Notices of the American Mathematical Society 'The book contains many exercises and three appendices presenting the material from analysis, probability and number theory that is used. Certainly the book is a good read for a mathematicians interested in the interaction between probability theory and number theory. The techniques used in the book appear quite advanced to us, so we would recommend the book for students at a graduate but not at an undergraduate level.' Jörg Neunhäuserer, Mathematical Reviews 'The book is very well written - as expected by an author who has already contributed very widely used and important books - and certainly belongs to all libraries of universities and research institutes. It has all the attributes to make a classic textbook in this fascinating domain.' Michael Th. Rassias, zbMATH

Muu info

This introductory textbook for graduate students presents modern developments in probabilistic number theory, many for the first time.
Preface xi
Prerequisites and Notation xiii
1 Introduction
1(25)
1.1 Presentation
1(1)
1.2 How Does Probability Link with Number Theory Really?
2(2)
1.3 A Prototype: Integers in Arithmetic Progressions
4(11)
1.4 Another Prototype: The Distribution of the Euler Function
15(6)
1.5 Generalizations
21(3)
1.6 Outline of the Book
24(2)
2 Classical Probabilistic Number Theory
26(21)
2.1 Introduction
26(1)
2.2 Distribution of Arithmetic Functions
26(5)
2.3 The Erdos--Kac Theorem
31(8)
2.4 Convergence without Renormalization
39(6)
2.5 Final Remarks
45(2)
3 The Distribution Of Values Of The Riemann Zeta Function, I
47(28)
3.1 Introduction
47(3)
3.2 The Theorems of Bohr--Jessen and of Bagchi
50(17)
3.3 The Support of Bagchi's Measure
67(5)
3.4 Generalizations
72(3)
4 The Distribution Of Values Of The Riemann Zeta Function, II
75(22)
4.1 Introduction
75(1)
4.2 Strategy of the Proof of Selberg's Theorem
76(9)
4.3 Dirichlet Polynomial Approximation
85(4)
4.4 Euler Product Approximation
89(6)
4.5 Further Topics
95(2)
5 The Chebychev Bias
97(27)
5.1 Introduction
97(1)
5.2 The Rubinstein-Sarnak Distribution
98(4)
5.3 Existence of the Rubinstein-Sarnak Distribution
102(11)
5.4 The Generalized Simplicity Hypothesis
113(10)
5.5 Further Results
123(1)
6 The Shape Of Exponential Sums
124(20)
6.1 Introduction
124(5)
6.2 Proof of the Distribution Theorem
129(10)
6.3 Applications
139(4)
6.4 Generalizations
143(1)
7 Further Topics
144(13)
7.1 Equidistribution Modulo 1
144(4)
7.2 Roots of Polynomial Congruences and the Chinese Remainder Theorem
148(3)
7.3 Gaps between Primes
151(1)
7.4 Cohen--Lenstra Heuristics
152(1)
7.5 Ratner Theory
153(2)
7.6 And Even More
155(2)
Appendix A Analysis
157(14)
A.1 Summation by Parts
157(1)
A.2 The Logarithm
158(1)
A.3 Mellin Transform
159(3)
A.4 Dirichlet Series
162(5)
A.5 Density of Certain Sets of Holomorphic Functions
167(4)
Appendix B Probability
171(51)
B.1 The Riesz Representation Theorem
171(1)
B.2 Support of a Measure
172(2)
B.3 Convergence in Law
174(3)
B.4 Perturbation and Convergence in Law
177(5)
B.5 Convergence in Law in a Finite-Dimensional Vector Space
182(5)
B.6 The Weyl Criterion
187(7)
B.7 Gaussian Random Variables
194(3)
B.8 Sub-Gaussian Random Variables
197(2)
B.9 Poisson Random Variables
199(1)
B.10 Random Series
200(9)
B.11 Some Probability in Banach Spaces
209(13)
Appendix C Number Theory
222(24)
C.1 Multiplicative Functions and Euler Products
222(4)
C.2 Additive Functions
226(1)
C.3 Primes and Their Distribution
227(3)
C.4 The Riemann Zeta Function
230(4)
C.5 Dirichlet L-Functions
234(6)
C.6 Exponential Sums
240(6)
References 246(6)
Index 252
Emmanuel Kowalski is Professor in the Mathematics Department of the Swiss Federal Institute of Technology, Zurich. He is the author of five previous books, including the widely cited Analytic Number Theory (2004) with H. Iwaniec, which is considered to be the standard graduate textbook for analytic number theory.