Course Outline and Some Class Notes

For the most part we shall (unashamedly) follow closely the appropriate sections of the references listed below.

• Introduction
• Prime Numbers

• Elementary Theory of Prime Numbers
• Infinitude of Primes
• Prime Number Estimates of Chebyshev
• Here is a simple proof of Bertrand's Postulate (we did not discuss this approach in class)
  
• Mertens' Identities (and another theorem of Chebyshev)
• Primes in Arithmetic Progressions - Dirichlet's Theorem
• Supplements
• Characters on Finite Abelian Groups
• Arithmetic Functions and Dirichlet Convolution

• Elementary Sieve Methods
• The Sieve of Eratosthenes-Legendre
• The Brun Sieve
• The Selberg Sieve
• Schnirelmann's Theorem
• Mann's Theorem (we did not need or cover this material in class)

• Halasz's Theorem and a "Pretentious" Proof of the Prime Number Theorem
• Relations Equivalent to the Prime Number Theorem (see also Chapter 1 of Ben Green's notes)
• Further Equivalences (we did not cover this material in class)
• Halasz's Theorem and the Prime Number Theorem (Chapter 2 of Ben Green's notes)
  
• Proof of Halasz's Theorem (Chapter 4 of Ben Green's notes)
  
• Supplements
• Asymptotics for the Mean Value of some Arithmetic Functions
• Some Fourier Analysis (Chapter 3 of Ben Green's notes)

• The Classical Proof of the Prime Number Theorem
• Riemann plan for proving the prime number theorem
• Properties if the Riemann-zeta function
• Perron's formula and the explicit formula
• The zero-free region and the prime number theorem
• Supplements
• The gamma function
• Integral functions of order one
  

• Newman's Short Proof of the Prime Number Theorem

• Dirichlet's Theorem (revisited)

• Prime Number Theorem for Primes in Arithmetic Progressions (we did not cover this in class, below are links to notes of Ben Green)
  
• Dirichlet Charaters
• Basic properties of Dirichlet L-functions
• Zero-free regions for Dirichlet L-functions I
• Zero-free regions for Dirichlet L-functions II
• The Siegel-Walfisz Theorem

• Vinogradov's Three Primes Theorem
• Goldbach Problems (Notes of Alex Rice from the Fall 2009 Analysis Learning Seminar)
• The Vaughan Identities
  

• Additional Notes
• An Elementary Proof of the Prime Number Theorem
  

Primary Resources:

• Not Always Buried Deep, by Paul Pollack (for the elementary theory and some of the sieve methods)
• Multiplicative Number Theory I. Classical Theory, by Hugh L. Montgomery and Robert C. Vaughan 
• Prime Numbers, course notes of Ben Green

Secondary Resources:

• Analytic Number Theory, by Henryk Iwaniec and Emmanuel Kowalski
• Multiplicative Number Theory, by Harold Davenport
• Prime Numbers and Their Distribution, by Gerald Tenenbaum and Michel Mendes France
• Introduction to Analytic and Probabilistic Number Theory, by Gerald Tenenbaum
• Additive Number Theory - The Classical Bases, by Melvyn B. Nathanson
• Prime Numbers, course notes of Andrew Granville

Some Additional Elemental and Elementary Number Theory Resources:

• Elementary Methods in Number Theory, by Melvyn B. Nathanson
• Introduction to Analytic Number Theory, by Tom M. Apostol
• Number Theory, course notes of Pete L. Clark (versions of some individual sections can be found here)