This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression.
The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.
Readership: Final-year undergraduates and first-year graduates with basic knowledge of complex analysis and abstract algebra; academics.
Contents:
- Facts about Integers
- Arithmetical Functions
- Averages of Arithmetical Functions
- Elementary Results on the Distribution of Primes
- The Prime Number Theorem
- Dirichlet Series
- Primes in Arithmetic Progression
Author(s): Heng Huat Chan
Series: Monoghraphs in Number Theory 3
Publisher: World Scientific
Year: 2009
Language: English
Pages: 128
City: Singapore
1. The Fundamental Theorem of Arithmetic
1.1 Least Integer Axiom and Mathematical Induction
1.2 Division Algorithm
1.3 Greatest common divisors
1.4 The Euclidean Algorithm
1.5 Congruences
1.6 Fundamental Theorem of Arithmetic
1.7 Exercises
2. Arithmetical Functions and Dirichlet Multiplication
2.1 The Möbius function
2.2 The Euler totient function
2.3 Dirichlet product
2.4 Dirichlet inverses and the M¨obius inversion formula
2.5 Multiplicative functions and Dirichlet products
2.6 Exercises
3. Averages of Arithmetical Functions
3.1 Introduction
3.2 Partial summation and the Euler-Maclaurin summation formula
3.3 Some elementary asymptotic formulas
3.4 The divisor function and Dirichlet’s hyperbola method
3.5 An application of the hyperbola method
3.6 Exercises
4. Elementary Results on the Distribution of Primes
4.1 Introduction
4.2 The function ψ(x)
4.3 The functions θ(x) and π(x)
4.4 Merten’s estimates
4.5 Prime Number Theorem and M(μ)
4.6 The Bertrand Postulate
4.7 Exercises
5. The Prime Number Theorem
5.1 The Prime Number Theorem
5.2 The Riemann zeta function
5.3 Euler’s product and the product representation of ζ(s)
5.4 Analytic continuation of ζ(s) to σ > 0
5.5 Upper bounds for |ζ(s)| and |ζ′(s)| near σ = 1
5.6 The non-vanishing of ζ(1 + it)
5.7 A lower bound for |ζ(s)| near σ = 1
5.8 Perron’s Formula
5.9 Completion of the proof of the Prime Number Theorem
5.10 Exercises
6. Dirichlet Series
6.1 Absolute convergence of a Dirichlet series
6.2 The Uniqueness Theorem
6.3 Multiplication of Dirichlet series
6.4 Conditional convergence of Dirichlet series
6.5 Landau’s Theorem for Dirichlet series
6.6 Exercises
7. Primes in Arithmetic Progression
7.1 Introduction
7.2 Dirichlet’s characters
7.3 The orthogonal relations
7.4 The Dirichlet L-series
7.5 Proof of Dirichlet’s Theorem
7.6 Exercises
