## Content of Number Theory: An Introduction to Mathematics.

**Part A**

I The Expanding Universe of Numbers.

0 Sets, Relations and Mappings

1 Natural Numbers

2 Integers and Rational Numbers

3 Real Numbers

4 Metric Spaces

5 Complex Numbers

6 Quaternions and Octonions

7 Groups

8 Rings and Fields

9 Vector Spaces and Associative Algebras

10 Inner Product Spaces

11 Further Remarks

12 Selected References

Additional References

II Divisibility

1 Greatest Common Divisors

2 The B´ezout Identity

3 Polynomials

4 Euclidean Domains

5 Congruences

6 Sums of Squares

7 Further Remarks

8 Selected References

Additional References

viii Contents

III More on Divisibility

1 The Law of Quadratic Reciprocity

2 Quadratic Fields

3 Multiplicative Functions

4 Linear Diophantine Equations

5 Further Remarks

6 Selected References

Additional References

IV Continued Fractions and Their Uses

1 The Continued Fraction Algorithm

2 Diophantine Approximation

3 Periodic Continued Fractions

4 Quadratic Diophantine Equations

5 The Modular Group

6 Non-Euclidean Geometry

7 Complements

8 Further Remarks

9 Selected References

Additional References

V Hadamard’s Determinant Problem

1 What is a Determinant?

2 Hadamard Matrices

3 The Art of Weighing

4 Some Matrix Theory

5 Application to Hadamard’s Determinant Problem

6 Designs

7 Groups and Codes

8 Further Remarks

9 Selected References

VI Hensel’s p-adic Numbers

1 Valued Fields

2 Equivalence

3 Completions

4 Non-Archimedean Valued Fields

5 Hensel’s Lemma

6 Locally Compact Valued Fields

7 Further Remarks

8 Selected References

Contents ix

**Part B**

VII The Arithmetic of Quadratic Forms

1 Quadratic Spaces

2 The Hilbert Symbol

3 The Hasse–Minkowski Theorem

4 Supplements

5 Further Remarks

6 Selected References

VIII The Geometry of Numbers

1 Minkowski’s Lattice Point Theorem

2 Lattices

3 Proof of the Lattice Point Theorem; Other Results

4 Voronoi Cells

5 Densest Packings

6 Mahler’s Compactness Theorem

7 Further Remarks

8 Selected References

Additional References

IX The Number of Prime Numbers

1 Finding the Problem

2 Chebyshev’s Functions

3 Proof of the Prime Number Theorem

4 The Riemann Hypothesis

5 Generalizations and Analogues

6 Alternative Formulations

7 Some Further Problems

8 Further Remarks

9 Selected References

Additional References

X A Character Study

1 Primes in Arithmetic Progressions

2 Characters of Finite Abelian Groups

3 Proof of the Prime Number Theorem for Arithmetic Progressions

4 Representations of Arbitrary Finite Groups

5 Characters of Arbitrary Finite Groups

6 Induced Representations and Examples

7 Applications

8 Generalizations

9 Further Remarks

10 Selected References

x Contents

XI Uniform Distribution and Ergodic Theory

1 Uniform Distribution

2 Discrepancy

3 Birkhoff’s Ergodic Theorem

4 Applications

5 Recurrence

6 Further Remarks

7 Selected References

Additional Reference

XII Elliptic Functions

1 Elliptic Integrals

2 The Arithmetic-Geometric Mean

3 Elliptic Functions

4 Theta Functions

5 Jacobian Elliptic Functions .

6 The Modular Function

7 Further Remarks

8 Selected References

XIII Connections with Number Theory

1 Sums of Squares

2 Partitions

3 Cubic Curves

4 Mordell’s Theorem

5 Further Results and Conjectures

6 Some Applications

7 Further Remarks

8 Selected References

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