Copies of the classnotes are on the internet in PDF format as given below. The "Examples, Exercises, and Proofs" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Examples, Exercises, and Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

These notes can be used as supplements to the ETSU class Theory of Numbers (MATH 5070). The catalog description for Theory of Numbers in the 2014-15 graduate catalog is: "Divisibility, congruences, quadratic residues, Diophantine equations, and a brief treatment of binary quadratic forms." However, that course was removed from the catalog in 2015. The 2014-15 ETSU graduate catalog can be found in the ETSU Catalog Archives (accessed 3/7/2022). A better description of that class would be: "Unique factrization, congruence, quadratic/cubic/biquadratic reciprocity, finite fields, the zeta function." Some online notes for Theory of Numbers are available.

1. Chapter 1. Foundations.
2. Chapter 2. Some Important Dirichlet Series and Arithmetic Functions.
3. Chapter 3. The Basic Theorems.
4. Chapter 4. Prime Numbers in Residue Classes: Dirichlet's Theorem.
5. Chapter 5. Error Estimates and the Riemann Hypothesis.
6. Chapter 6. An "Elementary" Proof of the Prime Number Theorem.
7. Appendices.

Preface. Preface notes

Chapter 1. Foundations.

• Section 1.1. Counting Prime Numbers. Section 1.1 notes
  • Supplement. Exercises and Proofs from Section 1.1. Beamer file of Section 1.1 proofs
  • Supplement. Printouts of Exercises and Proofs from Section 1.1. Print file of Section 1.1 proofs
• Section 1.2. Arithmetic Functions. Section 1.2 notes
  • Supplement. Exercises and Proofs from Section 1.2. Beamer file of Section 1.2 proofs
  • Supplement. Printouts of Exercises and Proofs from Section 1.2. Print file of Section 1.2 proofs
• Section 1.3. Abel Summation (partial). Section 1.3 notes
• Section 1.4. Estimation of Sums by Integrals; Euler's Summation Formula.
• Section 1.5. The Function li(x).
• Section 1.6. Chebyshev's Theta Function.
• Section 1.7. Dirichlet Series and the Zeta Function.
• Section 1.8. Convolutions.
• Study Guide 1.

Chapter 2. Some Important Dirichlet Series and Arithmetic Functions.

• Section 2.1. The Euler Product.
• Section 2.2. The Möbius Function.
• Section 2.3.The Series for log ζ(s) and ζ'(s)/ζ(s).
• Section 2.4. Chebyshev's Psi Function and Powers of Primes.
• Section 2.5. Estimates of Some Summation Functions.
• Section 2.6. Mertens's Estimates.
• Study Guide 2.

Chapter 3. The Basic Theorems.

• Section 3.1. Extension of the Definition of the Zeta Function.
• Section 3.2. Inversion of Dirichlet Series; The Integral Version of the Fundamental Theorem.
• Section 3.3. An Alternative Method: Newman's Proof.
• Section 3.4. The Limit and Series Versions of the Fundamental Theorem; The Prime Number Theorem.
• Section 3.5. Some Applications of the Prime Number Theorem.
• Study Guide 3.

Chapter 4. Prime Numbers in Residue Classes: Dirichlet's Theorem.

• Section 4.1. Characters of Finite Abelian Groups.
• Section 4.2. Dirichlet Characters.
• Section 4.3. Dirichlet L-Functions.
• Section 4.4. Prime Numbers in Residue Classes.
• Study Guide 4.

Chapter 5. Error Estimates and the Riemann Hypothesis.

• Section 5.1. Error Estimates.
• Section 5.2. Connections with the Riemann Hypothesis.
• Section 5.3. The Zero-Free Region of the Zeta Function.
• Study Guide 5.

Chapter 6. An "Elementary" Proof of the Prime Number Theorem.

• Section 6.1. Framework of the Proof.
• Section 6.2. Selberg's Formulae and Completion of the Proof.
• Study Guide 6.

Appendices.

• Appendix A. Complex Functions of a Real Variable.
• Appendix B. Double Series and Multiplication of Series.
• Appendix C. Infinite Products.
• Appendix D. Differentiation Under the Integral Sign.
• Appendix E. The O, o Notation.
• Appendix F. Computing Values of π(x).
• Appendix G. Table of Primes.
• Appendix H. Biographical Notes.

Return to Bob Gardner's home page