Complex Analysis with Applications to Number Theory 9811590966, 9789811590962

The book discusses major topics in complex analysis with applications to number theory. This book is intended as a text

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English Pages 287 [297] Year 2020

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Complex Analysis with Applications to Number Theory
 9811590966, 9789811590962

Table of contents :
Preface
Contents
About the Author
Symbols
1 Introduction and Simply Connected Regions
1.1 Introduction
1.2 Connectedness
1.3 Extended Complex Plane
1.4 Distance Between Non-intersecting Compact and Closed sets
1.5 Complex Integral
1.6 Homotopic Paths
1.7 Results on Simply Connected Regions
1.8 Notation for Denoting Constants
1.9 Exercises
2 The Cauchy Theorems and Their Applications
2.1 Introduction
2.2 The Cauchy Theorem and the Cauchy Integral Formula for Convex Open Sets
2.3 An Account of Some Basic Results on Analytic Functions
2.4 The Cauchy Theorem for Closed Paths in A Simply Connected Region
2.5 The Cauchy Integral Formula and the Cauchy Theorem for Cycles Homologous To Zero in an Open Set and the Cauchy Residue Theorem
2.6 Argument Principle, Open Mapping Theorem, Maximum Modulus Principle, the Rouché Theorem and The Jensen Formula
2.7 The Phragmen–Lindelöf Method: Maximum Modulus Principle in an Unbounded Strip and the Hadamard Three-Circle Theorem
2.8 Existence of an Analytic Branch of Logarithm of Non-vanishing Analytic Functions
2.9 The Jensen Formula
2.10 An Estimate for the Number of Zeros of an Exponential Polynomial in a Disc
2.11 Exercises
3 Conformal Mappings and the Riemann Mapping Theorem
3.1 Introduction
3.2 Examples of Explicit Conformal Mappings
3.3 Automorphisms of the Open Unit Disc
3.4 Automorphisms of the Upper Half Plane
3.5 The Riemann Mapping Theorem and More General Theorem 3.11
3.6 Lemmas for the Proof of Theorem 3.11
3.7 Proof of Theorem 3.11
3.8 Exercises
4 Harmonic Functions
4.1 Introduction
4.2 The Cauchy–Riemann Equations
4.3 Definition and Examples of Harmonic Functions
4.4 Harmonic Conjugate of a Harmonic Function in a Simply Connected Region
4.5 Maximum Principle for Harmonic Functions Satisfying MVP
4.6 The Dirichlet Problem for Open Discs
4.7 Exercises
5 The Picard Theorems
5.1 Introduction
5.2 The Borel and Carathéodory Lemma and Other Results for the Picard Theorems
5.3 Proof of the Schottky Theorem 5.2
5.4 Proofs of the Little Picard Theorem 5.1 and the Great Picard Theorem 5.3
5.5 Exercises
6 The Weierstrass Factorisation Theorem, Hadamard's Factorisation Theorem and the Gamma Function
6.1 Introduction
6.2 Infinite Products
6.3 The Weierstrass Elementary Factors
6.4 The Weierstrass Factorisation Theorem
6.5 Hadamard's Factorisation Theorem
6.6 Extension of the Weierstrass Factorisation Theorem for an Arbitrary Region
6.7 Representation of Meromorphic Functions by Partial Fractions
6.8 Applications of the Weierstrass Factorisation Theorem
6.9 The Gamma Function
6.10 Integral Representation for Γ(z)
6.11 The Bernoulli Numbers and the Bernoulli Polynomials
6.12 The Euler–Maclaurin–Jacobi Sum Formula
6.13 The Stirling Formula
6.14 The Beta Function
6.15 Exercises
7 The Riemann Zeta Function and the Prime Number Theorem
7.1 Introduction
7.2 The Euler Product for ζ(s)
7.3 Applications of the Euler Product for ζ(s)
7.4 The Abel Summation Formula and Integral Representations for ζ'(s)ζ(s)
7.5 Analytic Continuation of ζ(s) in σ>0 and Its Non-vanishing on the Line σ=1
7.6 Estimates for ζ(s) and ζ'(s)
7.7 Introduction to the Prime Number Theorem
7.8 Equivalence of PNT and the Non-vanishing of ζ(s) on the Line σ=1
7.9 Lemmas for the Proof of the Wiener–Ikehara Theorem 7.18
7.10 Proof of Theorem 7.18
7.11 Analytic Continuation of ζ(s) in C and Functional Equation for ζ(s)
7.12 The Function µ(σ)
7.13 Main Conjectures in the Theory of the Riemann Zeta Function
7.14 Exercises
8 The Prime Number Theorem with an Error Term
8.1 Introduction
8.2 Positive Lower Bound for |ζ(1+it)| and Zero-Free Region for ζ(s)
8.3 An Equivalent Version of Prime Number Theorem …
8.4 Integral Representation for ψ1(x)
8.5 Proof of Theorem 8.1
8.6 Exercises
9 The Dirichlet Series and the Dirichlet Theorem on Primes in Arithmetic Progressions
9.1 Introduction
9.2 Convergence of the Dirichlet Series
9.3 Multiplication of Two Dirichlet Series
9.4 Another Proof of Theorem 7.9摥映數爠eflinkthm6.77.97 That ζ(1+it)neq0
9.5 Characters of Finite Abelian Groups
9.5.1 Properties of Characters
9.6 The Dirichlet Characters
9.7 The Dirichlet L-Functions
9.8 Non-vanishing of L(1, χ)
9.9 The Dirichlet Theorem on Primes in Arithmetic Progression
9.10 The Prime Number Theorem in an Arithmetic Progression
9.11 Exercises
10 The Baker Theorem
10.1 Introduction
10.2 The Thue-Siegel Lemma
10.3 Proof of the Baker Theorem 10.1
10.4 Proof of Corollary 10.2
10.5 An Application of (10.1.1)
10.6 Exercises
Appendix References
Index

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