Arithmetic Tales - Advanced Edition

Preface
Preliminaries
1 General Notation
2 Basics of Complex Analysis
3 Functions
3.1 The Zeta and Gamma Functions
3.2 Integer Functions
3.3 Sums and Products
3.4 Exponential and Logarithmic Functions
3.5 Comparison Relations
Contents
1 Basic Tools
1.1 The Riemann–Stieltjes Integral
1.1.1 Definition
1.1.2 Basic Properties
1.1.3 Integration by Parts
1.1.4 Sums and Integrals
1.2 Partial Summation
1.2.1 Abel Transformation Formula
1.2.2 Partial Summation Formula
1.3 The Euler–Maclaurin Summation Formula
Exercises
References
2 Linear Diophantine Equations
2.1 Basic Facts
2.1.1 Solutions in the Simplest Cases
2.1.2 The Frobenius Problem
2.2 The Ring (Z /n Z , + , )
2.2.1 Units and Zero Divisors
2.2.2 The Euler Totient Function
2.2.3 The Euler–Fermat Theorem
2.3 Denumerants
2.3.1 Definition
2.3.2 Denumerants with Two Variables
2.3.2.1 Paoli's Theorem
2.3.2.2 Ehrhart's Theorem
2.3.2.3 Tripathi's Theorem
2.3.3 Denumerants with k Variables
2.3.3.1 Bounds
2.3.3.2 Asymptotic Formula
2.3.4 Generating Functions
2.3.5 The Barnes Zeta Function
Exercises
References
3 Prime Numbers
3.1 Primitive Roots
3.1.1 Multiplicative Order
3.1.2 Primitive Roots
3.1.2.1 Introduction
3.1.2.2 Basic Facts
3.1.2.3 Primitive Roots Modulo a Prime
3.1.3 Artin's Conjecture
3.1.3.1 Genesis of the Conjecture
3.1.3.2 Heuristics
3.1.3.3 Further Developments
3.1.4 Power Residues
3.2 Elementary Prime Numbers Estimates
3.2.1 Chebyshev's Functions of Primes
3.2.2 Chebyshev's Estimates
3.2.2.1 The Functions θ and ψ
3.2.2.2 The Functions π and pn
3.2.3 An Alternative Approach
3.2.4 Mertens's Theorems
3.3 The Riemann Zeta-Function
3.3.1 Euler, Dirichlet and Riemann
3.3.2 The Gamma and Theta Functions
3.3.2.1 Functional Equation
3.3.2.2 Complex Stirling's Formula
3.3.2.3 The Theta Function
3.3.3 Functional Equation
3.3.4 Approximate Functional Equations
3.3.5 Estimates For |ζ(s)|
3.3.6 Convexity Bounds
3.3.7 A Zero-Free Region
3.3.8 An Improved Zero-Free Region
3.3.9 The Resonance Method
3.4 Dirichlet L-Functions
3.4.1 Euclid vs. Euler
3.4.2 Dirichlet Characters
3.4.3 Dirichlet L-Functions
3.4.4 The Series p χ(p) p-1
3.4.5 The Non-vanishing of L(1,χ)
3.4.5.1 Complex Characters
3.4.5.2 Primitive Characters
3.4.5.3 Quadratic Characters
3.4.6 Functional Equation
3.5 The Prime Number Theorem
3.5.1 Perron Summation Formula
3.5.2 The Prime Number Theorem
3.5.2.1 The Classical Form
3.5.2.2 The Korobov–Vinogradov Zero-Free Region
3.5.2.3 Some Applications
3.5.2.4 Landau's Explicit Formula
3.5.3 Counting the Non-trivial Zeros
3.5.3.1 The Riemann–von Mangoldt Formula
3.5.3.2 The Density Hypothesis
3.5.4 The Siegel–Walfisz Theorem
3.5.4.1 Main Result
3.5.4.2 A Zero-Free Region for L-Functions
3.5.5 Explicit Estimates
3.5.5.1 L-Functions
3.5.5.2 Functions of Primes
3.6 The Riemann Hypothesis
3.6.1 The Genesis of the Conjecture
3.6.2 Hardy's Theorem
3.6.3 Some Consequences of the Riemann Hypothesis
3.6.3.1 Sharpening the Prime Number Theorem
3.6.3.2 Mertens' Conjecture
3.6.3.3 Arithmetic Functions
3.6.3.4 Redheffer's Matrix
3.6.3.5 Riesz's Criterion
3.6.3.6 Báez-Duarte Convolution
3.6.3.7 The Cyclotomic Polynomials
3.6.3.8 The Jensen Polynomials
3.6.3.9 An Integral
Exercises
References
4 Arithmetic Functions
4.1 The Basic Theory
4.1.1 The Ring of Arithmetic Functions
4.1.1.1 Definition
4.1.1.2 Fundamental Arithmetic Functions
4.1.1.3 Divisor Functions
4.1.1.4 Totients
4.1.2 Additive and Multiplicative Functions
4.1.2.1 Definitions
4.1.2.2 A Useful Criterion
4.1.2.3 Fundamental Examples
4.1.3 The Dirichlet Convolution Product
4.1.3.1 Definition
4.1.3.2 The Algebraic Point of View
4.1.3.3 Convolution Product and Multiplicativity
4.1.3.4 Fundamental Examples
4.1.4 The Möbius Inversion Formula
4.1.5 The Dirichlet Hyperbola Principle
4.2 Dirichlet Series
4.2.1 The Formal Viewpoint
4.2.2 Absolute Convergence
4.2.3 Conditional Convergence
4.2.4 Analytic Properties
4.2.5 Multiplicative Aspects
4.3 General Mean Value Results
4.3.1 A Useful Upper Bound
4.3.1.1 The Main Result
4.3.1.2 The Wirsing Conditions
4.3.1.3 Auxiliary Lemmas
4.3.1.4 Proof of Theorem 4.9
4.3.2 A Simple Asymptotic Formula
4.3.2.1 Main Result
4.3.2.2 Lemmas
4.3.2.3 Proof of Theorem 4.10
4.3.2.4 Some Applications
4.3.3 Vinogradov's Lemma
4.3.4 Wirsing and Halász Results
4.3.5 The Selberg–Delange Method
4.3.6 Logarithmic Mean Values
4.3.7 Using the Functional Equation
4.3.8 Lower Bounds
4.3.9 Short Sums
4.3.9.1 Upper Bounds
4.3.9.2 Asymptotic Formulæ
4.3.10 Sub-multiplicative Functions
4.3.11 Additive Functions
4.3.12 Refinements
4.4 Usual Multiplicative Functions
4.4.1 The Möbius Function
4.4.2 Distribution of k-free Numbers
4.4.3 The Number of Divisors
4.4.4 Totients
4.4.4.1 Euler's and Jordan's Totients
4.4.4.2 Dedekind's Totient
4.4.5 The Sum of Divisors
4.4.6 The Hooley Divisor Function
4.4.6.1 Background
4.4.6.2 Hooley's Kernel
4.4.6.3 The Method of the Differential Inequality
4.4.6.4 The Generalization of Daniel and Brüdern
4.4.6.5 The Generalization of de la Bretèche and Tenenbaum
4.5 Arithmetic Functions of Several Variables
4.5.1 Definitions
4.5.2 Dirichlet Convolution
4.5.3 Dirichlet Convolute
4.5.4 Dirichlet Series
4.5.5 Mean Values
4.6 Sieves
4.6.1 Combinatorial Sieve
4.6.2 The Selberg's Sieve
4.6.3 The Large Sieve
4.6.3.1 Introduction
4.6.3.2 Additive Large Sieve
4.6.3.3 Arithmetic Large Sieve
4.6.3.4 Multiplicative Large Sieve
4.7 Selected Problems in Multiplicative Number Theory
4.7.1 Squarefree Values of n2+1
4.7.2 The Bombieri–Vinogradov Theorem
4.7.2.1 Introduction
4.7.2.2 Reduction to Primitive Characters
4.7.2.3 Vaughan's Mean Value Theorem
4.7.2.4 Completion of the Proof
4.7.2.5 Other Formulations
4.7.2.6 Extensions
4.7.3 Bounded Gaps Between Primes
4.7.3.1 de Polignac's Conjecture
4.7.3.2 Admissible Tuples
4.7.3.3 The GPY Method
4.7.3.4 Zhang's Breakthrough
4.7.4 The Titchmarsh Divisor Problem
4.7.5 Power Means of the Riemann Zeta Function
4.7.5.1 The Mean Square
4.7.5.2 Higher Power Means
4.7.6 The Dirichlet–Piltz Divisor Problem
4.7.6.1 Introduction
4.7.6.2 Estimates Using Contour Integration
4.7.6.3 Estimates Using Voronoï's Formula
4.7.7 Multidimensional Divisor Problem
4.7.7.1 Introduction
4.7.7.2 A ψ-Expression of ( a ; x )
4.7.7.3 A Survey of Estimates for ( a ; x )
4.7.7.4 Application to Some Number-Theoretic Problems
4.7.8 The Hardy–Ramanujan Inequality
4.7.8.1 Background
4.7.8.2 Generalization
4.7.9 Prime-Independent Multiplicative Functions
4.7.9.1 Definition
4.7.9.2 Local Density
4.7.9.3 Long Sums
4.7.9.4 Short Sums
4.7.10 Smooth Numbers
4.7.10.1 Definition
4.7.10.2 Bounds
Exercises
References
5 Lattice Points
5.1 Introduction
5.1.1 Multiplicative Functions over Short Segments
5.1.2 The Number R(f,N,δ)
5.1.3 Basic Lemmas
5.1.3.1 The Squarefree Number Problem
5.1.3.2 The r-Free Number Problem
5.1.3.3 The Square-Full Number Problem
5.1.4 Srinivasan's Optimization Lemma
5.1.5 Divided Differences
5.2 Criteria for Integer Points
5.2.1 The First Derivative Test
5.2.2 The Second Derivative Test
5.2.2.1 The Reduction Principle
5.2.2.2 The Main Result
5.2.3 The kth Derivative Test
5.3 The Theorem of Huxley and Sargos
5.3.1 Preparatory Lemmas
5.3.2 Major Arcs
5.3.3 The Proof of Theorem 5.5
5.3.4 Application
5.3.5 Refinements
5.4 The Method of Filaseta and Trifonov
5.4.1 Preparatory Lemma
5.4.2 Higher Divided Differences
5.4.3 Proof of the Main Result
5.4.4 Application
5.4.5 Generalization
5.5 Recent Results
5.5.1 Smooth Curves
5.5.2 Polynomials
Exercises
References
6 Exponential Sums
6.1 The ψ-Function
6.1.1 Back to the Divisor Problem
6.1.2 Vaaler's and Stec̆kin's Inequalities
6.2 Basic Inequalities
6.2.1 Cauchy–Schwarz
6.2.2 Weyl's Shift
6.2.3 Van der Corput's A-Process
6.3 Exponential Sums Estimates
6.3.1 The First Derivative Theorem
6.3.2 The Second Derivative Theorem
6.3.3 The Third Derivative Theorem
6.3.4 The kth Derivative Theorem
6.4 Applications to the ψ-Function
6.4.1 The First Derivative Test
6.4.2 The Second Derivative Test
6.4.3 The Third Derivative Test
6.4.4 The Dirichlet Divisor Problem
6.5 The Method of Exponent Pairs
6.5.1 Van der Corput's B-Process
6.5.2 Exponent Pairs
6.5.3 Applications
6.5.4 A New Third Derivative Theorem
6.6 Character Sums
6.6.1 Additive Characters and Gauss Sums
6.6.1.1 Additive Characters
6.6.1.2 Ramanujan Sum
6.6.1.3 Gauss Sums
6.6.2 Incomplete Character Sums
6.6.3 Kloosterman Sums
6.6.3.1 Definition
6.6.3.2 Weil's Bound
6.6.3.3 Incomplete Sums
6.6.3.4 Twisted Incomplete Sums
6.7 The Hardy–Littlewood Method
6.7.1 The Circle Method
6.7.1.1 Additive Prime Number Theory
6.7.1.2 Hardy, Littlewood and Ramanujan's Method
6.7.1.3 Vinogradov's Approach
6.7.1.4 The Major and Minor Arcs
6.7.1.5 Conclusion
6.7.2 The Discrete Circle Method
6.7.2.1 Step 1: Major and Minor Arcs
6.7.2.2 Step 2: Gauss Sums
6.7.2.3 Step 3: Poisson Summation
6.7.2.4 Step 4: The Large Sieve on Minor Arcs
6.7.2.5 Step 5: Semicubical Powers of Integers
6.7.2.6 Step 6: Final Step
6.7.2.7 Conclusion
6.8 Vinogradov's Method
6.8.1 Introduction
6.8.2 Vinogradov's Mean Value Theorem
6.8.3 Proving the Main Conjecture
6.8.4 Walfisz's Estimates
6.9 Vaughan's Identity
6.9.1 Introduction
6.9.2 Prime Numbers in Intervals
6.9.3 The von Mangoldt Function
6.9.4 The Möbius Function
6.9.5 Heath-Brown's Refinement
6.9.6 A Variant of Vaughan's Identity
6.10 The Chowla–Walum Conjecture
6.10.1 Genesis of the Conjecture
6.10.2 Related Results
6.11 Exponential Sums over a Finite Field
6.11.1 Main Result
6.11.2 An Overview of Mordell's Method
6.11.3 The Heart of Mordell's Method
6.11.4 Further Results
Exercises
References
7 Algebraic Number Fields
7.1 Introduction
7.2 Algebraic Numbers
7.2.1 Some Group-Theoretic Results
7.2.2 Polynomials
7.2.2.1 Irreducibility
7.2.2.2 Criteria for Irreducibility
7.2.2.3 Discriminant of a Polynomial
7.2.3 Algebraic Numbers
7.2.3.1 Field Extensions
7.2.3.2 Number Fields
7.2.4 The Ring of Integers
7.2.4.1 Algebraic Integers
7.2.4.2 The Ring OK
7.2.4.3 The Index of OK
7.2.5 Integral Bases
7.2.5.1 Norm and Trace
7.2.5.2 Integral Bases
7.2.5.3 Discriminants
7.2.6 Theorems for OK
7.2.6.1 Relations Between Discriminants
7.2.6.2 Stickelberger's and Kronecker's Theorems
7.2.6.3 A Formula with the Minimal Polynomial
7.2.6.4 Some Criteria for Integral Bases
7.2.6.5 Examples
7.2.7 Usual Number Fields
7.2.7.1 Quadratic Fields
7.2.7.2 Cyclotomic Fields
7.2.7.3 Pure Cubic Fields
7.2.7.4 Voronoï's Method for Cubic Fields
7.2.7.5 Pure Number Fields
7.2.7.6 Monogenic Number Fields
7.2.8 Units and Regulators
7.2.8.1 Dirichlet's Unit Theorem
7.2.8.2 Sketch of Proof of Theorem 7.5
7.2.8.3 Regulator
7.3 Ideal Theory
7.3.1 Arithmetic Properties
7.3.1.1 Prime Ideals
7.3.1.2 Dedekind Domains
7.3.2 Fractional Ideals
7.3.3 The Fundamental Theorem
7.3.4 Applications
7.3.4.1 Generators of Fractional Ideals
7.3.4.2 Unique Factorization
7.3.5 Norm of an Ideal
7.3.5.1 Definition
7.3.5.2 Basic Properties
7.3.5.3 Multiplicativity
7.3.6 Factorization of (p)
7.3.6.1 Primes Above Primes
7.3.6.2 Ramification
7.3.6.3 Galois Extensions
7.3.6.4 The Theorem of Kummer–Dedekind
7.3.7 Quadratic Fields
7.3.7.1 The Legendre–Jacobi–Kronecker Symbol
7.3.7.2 Prime Ideal Decomposition of (p)
7.3.7.3 The Conductor of a Quadratic Field
7.3.8 The Class Group
7.3.8.1 Definition
7.3.8.2 Finiteness of the Class Group
7.3.8.3 The Minkowski Bound
7.4 The Dedekind Zeta-Function
7.4.1 The Function rK
7.4.2 The Function ζK
7.4.3 Functional Equation
7.4.4 Explicit Convexity Bound
7.4.5 Subconvexity Bound
7.4.6 Zero-Free Region
7.4.7 Application to the Class Number
7.4.8 Lower Bounds for |dK|
7.4.9 Quadratic Fields
7.5 Selected Problems in Algebraic Number Theory
7.5.1 Computations of Galois Groups
7.5.1.1 Fundamental Tools
7.5.1.2 Resolvents
7.5.1.3 Usual Examples
7.5.2 Gauss's Class Number Problems
7.5.2.1 Euler's Polynomials
7.5.2.2 Class Number One Problems
7.5.2.3 Background of Gauss' Conjecture
7.5.2.4 Elliptic Curves
7.5.3 The Brauer–Siegel Theorem
7.5.4 The Class Number Formula
7.5.5 The Prime Ideal Theorem
7.5.6 The Ideal Theorem
7.5.6.1 Background
7.5.6.2 Main Results
7.5.7 The Kronecker–Weber Theorem
7.5.8 Class Field Theory Over Q
7.5.8.1 Background
7.5.8.2 Artin Map
7.5.8.3 Main Theorems
7.5.8.4 Characters
7.5.9 Primes of the Form x2+ny2
7.5.9.1 Places
7.5.9.2 Hilbert Class Field
7.5.9.3 A Particular Case
7.5.9.4 The General Case
7.5.10 The Chebotarëv's Density Theorem
7.5.10.1 Densities
7.5.10.2 Artin Symbol
7.5.10.3 Chebotarëv's Theorem
7.5.10.4 Dirichlet's Theorem
7.5.10.5 An Application
7.5.11 Artin L-Functions
7.5.11.1 An Overview of Class Field Theory
7.5.11.2 Ray Class Characters
7.5.11.3 Hecke L-Functions
7.5.11.4 Representation Theory of Finite Groups
7.5.11.5 Ramification
7.5.11.6 Artin L-Functions
7.5.11.7 Main Properties of Artin L-Functions
7.5.11.8 The Great Conjectures
7.5.11.9 Application
7.5.12 Analytic Methods
7.5.12.1 Ideal Classes
7.5.12.2 Regulator
Exercises
References
Hints and Answers to Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
References
Index