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Arithmetic Tales: Advanced Edition 2nd ed. 2020 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 782 pages, aukštis x plotis: 235x155 mm, weight: 1211 g, 5 Illustrations, color; 3 Illustrations, black and white; XIX, 782 p. 8 illus., 5 illus. in color., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 27-Nov-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030549453
  • ISBN-13: 9783030549459
Kitos knygos pagal šią temą:
Arithmetic Tales: Advanced Edition 2nd ed. 2020Didesnis paveikslėlis
  • Formatas: Paperback / softback, 782 pages, aukštis x plotis: 235x155 mm, weight: 1211 g, 5 Illustrations, color; 3 Illustrations, black and white; XIX, 782 p. 8 illus., 5 illus. in color., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 27-Nov-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030549453
  • ISBN-13: 9783030549459
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030549459.html
This textbook covers a wide array of topics in analytic and multiplicative number theory, suitable for graduate level courses.Extensively revised and extended, this Advanced Edition takes a deeper dive into the subject, with the elementary topics of the previous edition making way for a fuller treatment of more advanced topics. The core themes of the distribution of prime numbers, arithmetic functions, lattice points, exponential sums and number fields now contain many more details and additional topics. In addition to covering a range of classical and standard results, some recent work on a variety of topics is discussed in the book, including arithmetic functions of several variables, bounded gaps between prime numbers ą la Yitang Zhang, Mordell's method for exponential sums over finite fields, the resonance method for the Riemann zeta function, the Hooley divisor function, and many others. Throughout the book, the emphasis is on explicit results.

Assuming only familiarity with elementary number theory and analysis at an undergraduate level, this textbook provides an accessible gateway to a rich and active area of number theory. With an abundance of new topics and 50% more exercises, all with solutions, it is now an even better guide for independent study.

Recenzijos

Long chapters with many (exciting) subsections. There are a lot of problems to do (and, yes, do them); there are hints and answers in the back. The book is very well-written, doing a lot of deep stuff in detail and continually providing discussions of recent results, open questions, and so on. There are fantastic lists of references after each chapter. ... Its a wonderful book. (Michael Berg, MAA Reviews, January 30, 2022)

Preliminaries vii
1 General Notation vii
2 Basics of Complex Analysis vii
3 Functions viii
3.1 The Zeta and Gamma Functions viii
3.2 Integer Functions ix
3.3 Sums and Products x
3.4 Exponential and Logarithmic Functions x
3.5 Comparison Relations xi
1 Basic Tools
1(18)
1.1 The Riemann--Stieltjes Integral
1(7)
1.1.1 Definition
1(1)
1.1.2 Basic Properties
2(3)
1.1.3 Integration by Parts
5(2)
1.1.4 Sums and Integrals
7(1)
1.2 Partial Summation
8(2)
1.2.1 Abel Transformation Formula
8(1)
1.2.2 Partial Summation Formula
9(1)
1.3 The Euler-Maclaurin Summation Formula
10(5)
Exercises
15(3)
References
18(1)
2 Linear Diophantine Equations
19(26)
2.1 Basic Facts
19(7)
2.1.1 Solutions in the Simplest Cases
19(2)
2.1.2 The Frobenius Problem
21(5)
2.2 The Ring (Z/nZ, +, ×)
26(2)
2.2.1 Units and Zero Divisors
26(1)
2.2.2 The Euler Totient Function
27(1)
2.2.3 The Euler-Fermat Theorem
27(1)
2.3 Denumerants
28(11)
2.3.1 Definition
28(1)
2.3.2 Denumerants with Two Variables
28(4)
2.3.3 Denumerants with k Variables
32(3)
2.3.4 Generating Functions
35(3)
2.3.5 The Barnes Zeta Function
38(1)
Exercises
39(3)
References
42(3)
3 Prime Numbers
45(114)
3.1 Primitive Roots
46(8)
3.1.1 Multiplicative Order
46(1)
3.1.2 Primitive Roots
47(2)
3.1.3 Artin's Conjecture
49(3)
3.1.4 Power Residues
52(2)
3.2 Elementary Prime Numbers Estimates
54(15)
3.2.1 Chebyshev's Functions of Primes
55(2)
3.2.2 Chebyshev's Estimates
57(6)
3.2.3 An Alternative Approach
63(1)
3.2.4 Mertens's Theorems
64(5)
3.3 The Riemann Zeta-Function
69(28)
3.3.1 Euler, Dirichlet and Riemann
69(2)
3.3.2 The Gamma and Theta Functions
71(4)
3.3.3 Functional Equation
75(3)
3.3.4 Approximate Functional Equations
78(2)
3.3.5 Estimates For |ξ(s)|
80(1)
3.3.6 Convexity Bounds
81(3)
3.3.7 A Zero-Free Region
84(3)
3.3.8 An Improved Zero-Free Region
87(6)
3.3.9 The Resonance Method
93(4)
3.4 Dirichlet L-Functions
97(19)
3.4.1 Euclid vs. Euler
97(3)
3.4.2 Dirichlet Characters
100(3)
3.4.3 Dirichlet L-Functions
103(1)
3.4.4 The Series Σp Χ(p)P-1
104(3)
3.4.5 The Non-vanishing of L(1, Χ)
107(5)
3.4.6 Functional Equation
112(4)
3.5 The Prime Number Theorem
116(20)
3.5.1 Perron Summation Formula
116(2)
3.5.2 The Prime Number Theorem
118(7)
3.5.3 Counting the Non-trivial Zeros
125(2)
3.5.4 The Siegel--Walfisz Theorem
127(4)
3.5.5 Explicit Estimates
131(5)
3.6 The Riemann Hypothesis
136(15)
3.6.1 The Genesis of the Conjecture
136(2)
3.6.2 Hardy's Theorem
138(2)
3.6.3 Some Consequences of the Riemann Hypothesis
140(11)
Exercises
151(3)
References
154(5)
4 Arithmetic Functions
159(196)
4.1 The Basic Theory
159(22)
4.1.1 The Ring of Arithmetic Functions
159(3)
4.1.2 Additive and Multiplicative Functions
162(5)
4.1.3 The Dirichlet Convolution Product
167(6)
4.1.4 The Mobius Inversion Formula
173(3)
4.1.5 The Dirichlet Hyperbola Principle
176(5)
4.2 Dirichlet Series
181(18)
4.2.1 The Formal Viewpoint
181(1)
4.2.2 Absolute Convergence
182(4)
4.2.3 Conditional Convergence
186(3)
4.2.4 Analytic Properties
189(6)
4.2.5 Multiplicative Aspects
195(4)
4.3 General Mean Value Results
199(34)
4.3.1 A Useful Upper Bound
199(8)
4.3.2 A Simple Asymptotic Formula
207(4)
4.3.3 Vinogradov's Lemma
211(2)
4.3.4 Wirsing and Halasz Results
213(2)
4.3.5 The Selberg--Delange Method
215(2)
4.3.6 Logarithmic Mean Values
217(2)
4.3.7 Using the Functional Equation
219(2)
4.3.8 Lower Bounds
221(2)
4.3.9 Short Sums
223(5)
4.3.10 Sub-multiplicative Functions
228(1)
4.3.11 Additive Functions
229(2)
4.3.12 Refinements
231(2)
4.4 Usual Multiplicative Functions
233(20)
4.4.1 The Mobius Function
233(3)
4.4.2 Distribution of k-free Numbers
236(2)
4.4.3 The Number of Divisors
238(2)
4.4.4 Totients
240(3)
4.4.5 The Sum of Divisors
243(1)
4.4.6 The Hooley Divisor Function
244(9)
4.5 Arithmetic Functions of Several Variables
253(8)
4.5.1 Definitions
253(2)
4.5.2 Dirichlet Convolution
255(1)
4.5.3 Dirichlet Convolute
256(1)
4.5.4 Dirichlet Series
257(1)
4.5.5 Mean Values
258(3)
4.6 Sieves
261(21)
4.6.1 Combinatorial Sieve
261(5)
4.6.2 The Selberg's Sieve
266(8)
4.6.3 The Large Sieve
274(8)
4.7 Selected Problems in Multiplicative Number Theory
282(54)
4.7.1 Squarefree Values of n2 + 1
282(4)
4.7.2 The Bombieri--Vinogradov Theorem
286(9)
4.7.3 Bounded Gaps Between Primes
295(4)
4.7.4 The Titchmarsh Divisor Problem
299(4)
4.7.5 Power Means of the Riemann Zeta Function
303(4)
4.7.6 The Dirichlet--Piltz Divisor Problem
307(8)
4.7.7 Multidimensional Divisor Problem
315(9)
4.7.8 The Hardy--Ramanujan Inequality
324(2)
4.7.9 Prime-Independent Multiplicative Functions
326(6)
4.7.10 Smooth Numbers
332(4)
Exercises
336(13)
References
349(6)
5 Lattice Points
355(56)
5.1 Introduction
355(9)
5.1.1 Multiplicative Functions over Short Segments
355(3)
5.1.2 The Number R(f, N, δ)
358(1)
5.1.3 Basic Lemmas
359(3)
5.1.4 Srinivasan's Optimization Lemma
362(1)
5.1.5 Divided Differences
363(1)
5.2 Criteria for Integer Points
364(9)
5.2.1 The First Derivative Test
364(2)
5.2.2 The Second Derivative Test
366(4)
5.2.3 The kth Derivative Test
370(3)
5.3 The Theorem of Huxley and Sargos
373(18)
5.3.1 Preparatory Lemmas
374(2)
5.3.2 Major Arcs
376(6)
5.3.3 The Proof of Theorem 5.5
382(2)
5.3.4 Application
384(1)
5.3.5 Refinements
384(7)
5.4 The Method of Filaseta and Trifonov
391(11)
5.4.1 Preparatory Lemma
392(2)
5.4.2 Higher Divided Differences
394(3)
5.4.3 Proof of the Main Result
397(2)
5.4.4 Application
399(1)
5.4.5 Generalization
400(2)
5.5 Recent Results
402(2)
5.5.1 Smooth Curves
402(1)
5.5.2 Polynomials
403(1)
Exercises
404(4)
References
408(3)
6 Exponential Sums
411(106)
6.1 The Ψ-Function
411(5)
6.1.1 Back to the Divisor Problem
411(2)
6.1.2 Vaaler's and Steckin's Inequalities
413(3)
6.2 Basic Inequalities
416(4)
6.2.1 Cauchy--Schwarz
416(1)
6.2.2 Weyl's Shift
417(1)
6.2.3 Van der Corput's A-Process
418(2)
6.3 Exponential Sums Estimates
420(10)
6.3.1 The First Derivative Theorem
420(3)
6.3.2 The Second Derivative Theorem
423(4)
6.3.3 The Third Derivative Theorem
427(2)
6.3.4 The kth Derivative Theorem
429(1)
6.4 Applications to the k-Function
430(4)
6.4.1 The First Derivative Test
430(1)
6.4.2 The Second Derivative Test
431(1)
6.4.3 The Third Derivative Test
432(1)
6.4.4 The Dirichlet Divisor Problem
433(1)
6.5 The Method of Exponent Pairs
434(12)
6.5.1 Van der Corput's B-Process
434(5)
6.5.2 Exponent Pairs
439(3)
6.5.3 Applications
442(2)
6.5.4 A New Third Derivative Theorem
444(2)
6.6 Character Sums
446(9)
6.6.1 Additive Characters and Gauss Sums
446(4)
6.6.2 Incomplete Character Sums
450(2)
6.6.3 Kloosterman Sums
452(3)
6.7 The Hardy--Littlewood Method
455(17)
6.7.1 The Circle Method
455(9)
6.7.2 The Discrete Circle Method
464(8)
6.8 Vinogradov's Method
472(8)
6.8.1 Introduction
472(2)
6.8.2 Vinogradov's Mean Value Theorem
474(3)
6.8.3 Proving the Main Conjecture
477(1)
6.8.4 Walfisz's Estimates
478(2)
6.9 Vaughan's Identity
480(18)
6.9.1 Introduction
480(3)
6.9.2 Prime Numbers in Intervals
483(2)
6.9.3 The von Mangoldt Function
485(8)
6.9.4 The Mobius Function
493(1)
6.9.5 Heath-Brown's Refinement
494(4)
6.9.6 A Variant of Vaughan's Identity
498(1)
6.10 The Chowla--Walum Conjecture
498(2)
6.10.1 Genesis of the Conjecture
498(1)
6.10.2 Related Results
499(1)
6.11 Exponential Sums over a Finite Field
500(8)
6.11.1 Main Result
500(1)
6.11.2 An Overview of Mordell's Method
500(2)
6.11.3 The Heart of Mordell's Method
502(1)
6.11.4 Further Results
503(5)
Exercises
508(4)
References
512(5)
7 Algebraic Number Fields
517(158)
7.1 Introduction
517(1)
7.2 Algebraic Numbers
518(37)
7.2.1 Some Group-Theoretic Results
518(1)
7.2.2 Polynomials
519(5)
7.2.3 Algebraic Numbers
524(4)
7.2.4 The Ring of Integers
528(3)
7.2.5 Integral Bases
531(5)
7.2.6 Theorems for Ok
536(5)
7.2.7 Usual Number Fields
541(10)
7.2.8 Units and Regulators
551(4)
7.3 Ideal Theory
555(27)
7.3.1 Arithmetic Properties
555(3)
7.3.2 Fractional Ideals
558(3)
7.3.3 The Fundamental Theorem
561(2)
7.3.4 Applications
563(1)
7.3.5 Norm of an Ideal
564(4)
7.3.6 Factorization of (p)
568(6)
7.3.7 Quadratic Fields
574(3)
7.3.8 The Class Group
577(5)
7.4 The Dedekind Zeta-Function
582(16)
7.4.1 The Function r K
583(2)
7.4.2 The Function ξK
585(1)
7.4.3 Functional Equation
586(2)
7.4.4 Explicit Convexity Bound
588(1)
7.4.5 Subconvexity Bound
589(1)
7.4.6 Zero-Free Region
590(1)
7.4.7 Application to the Class Number
591(2)
7.4.8 Lower Bounds for |dK|
593(3)
7.4.9 Quadratic Fields
596(2)
7.5 Selected Problems in Algebraic Number Theory
598(68)
7.5.1 Computations of Galois Groups
598(4)
7.5.2 Gauss's Class Number Problems
602(4)
7.5.3 The Brauer--Siegel Theorem
606(3)
7.5.4 The Class Number Formula
609(1)
7.5.5 The Prime Ideal Theorem
610(2)
7.5.6 The Ideal Theorem
612(2)
7.5.7 The Kronecker-Weber Theorem
614(3)
7.5.8 Class Field Theory Over Q
617(6)
7.5.9 Primes of the Form x2 + ny2
623(9)
7.5.10 The Chebotarev's Density Theorem
632(4)
7.5.11 Artin L-Functions
636(22)
7.5.12 Analytic Methods
658(8)
Exercises
666(3)
References
669(6)
Hints and Answers to Exercises 675(102)
Index 777
Olivier Bordellčs has been an independent researcher in number theory for 15 years, essentially working in multiplicative number theory and the links between algebraic number theory and analytic number theory. Véronique Bordellčs is an English teacher who has translated several books in maths and science. She holds a BA in linguistics from the University of Manchester (UK) and also completed a Master's thesis in linguistics.