Analytic Number Theory for Beginners 2nd Edition by Prapanpong Pongsriiam ISBN 9781470464448 1470464446

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ISBN 10: 1470464446
ISBN 13: 9781470464448
Author: Prapanpong Pongsriiam

Analytic Number Theory for Beginners 2nd Edition Table of contents:

Chapter 1. Review of Elementary Number Theory
1.1. Divisibility
1.2. Greatest Common Divisor
1.3. Least Common Multiple
1.4. Prime Numbers
1.5. Congruences
1.6. Residue Systems
1.7. Chinese Remainder Theorem
1.8. Polynomial and Some Special Congruences
1.9. Lifting the Exponents
1.10. Primitive Roots
1.11. Quadratic Residues
1.12. Sum of Squares
1.13. Exercises
1.14. Notes

Chapter 2. Arithmetic Functions I
2.1. Introduction
2.2. Multiplicative Functions
2.3. Dirichlet Product
2.4. Divisor Sums and the Möbius Inversion Formula
2.5. Extensions of the Möbius Inversion Formula
2.6. Exercises
2.7. Notes

Chapter 3. The Floor Function
3.1. Introduction
3.2. Hermite’s Identity and Generalizations
3.3. Counting Lattice Points
3.4. Miscellaneous Examples
3.5. Exercises
3.6. Notes

Chapter 4. Summation Formulas
4.1. Introduction
4.2. Big O, Little o, and Related Notations
4.3. Sums of Monotone Functions
4.4. Partial Summation Formula
4.5. Review of the Riemann-Stieltjes Integral
4.6. Euler Summation Formula
4.7. Euler-Maclaurin Summation Formula
4.8. Exercises
4.9. Notes

Chapter 5. Arithmetic Functions II
5.1. Introduction
5.2. Average Orders
5.3. Dirichlet’s Hyperbola Method
5.4. Extremal Orders
5.5. Normal Orders
5.6. Exercises
5.7. Notes

Chapter 6. Elementary Results on the Distribution of Primes
6.1. Introduction
6.2. Chebyshev’s Estimates
6.3. Mertens’ Theorems
6.4. Exercises
6.5. Notes

Chapter 7. Characters and Dirichlet’s Theorem
7.1. Introduction
7.2. The Orthogonality Relations
7.3. Dirichlet Characters
7.4. Dirichlet L-Functions
7.5. Dirichlet’s Theorem
7.6. Applications of Dirichlet’s Theorem
7.7. The Statement of the Green-Tao Theorem
7.8. Exercises
7.9. Notes

Chapter 8. The Riemann Zeta Function
8.1. Introduction
8.2. Half-Plane of Convergence
8.3. Review of Infinite Products
8.4. Euler Product Formula
8.5. Analytic Continuation
8.6. Zero Free Regions
8.7. Exercises
8.8. Notes

Chapter 9. Prime Number Theorem and Some Extensions
9.1. Introduction
9.2. Proof of the Prime Number Theorem
9.3. Elementary Proof of the Prime Number Theorem
9.4. Stronger Forms of the Prime Number Theorem
9.5. Integers Having k Prime Factors
9.6. Exercises
9.7. Notes on Prime in Arithmetic Progressions
9.8. Analogues of the Bombieri-Vinogradov Theorem
9.9. Notes on Bounded Gaps Between Primes

Chapter 10. Introduction to Other Topics
10.1. Idea of Sieves
10.2. Brun’s Sieve
10.3. Selberg’s Sieve
10.4. Introduction to Additive Number Theory
10.5. Schnirelmann’s Basis Theorem
10.6. An Outline of the Circle Method in Waring’s Problem

Chapter 11. Hints for Selected Exercises

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Tags: Prapanpong Pongsriiam, Analytic Number, Beginners