10 Lectures on Analysis and Number Theory

Lecture 1:   Number Theory I - Basic Prime Number Theory
Lecture 2:   Number Theory II - The Riemann-Zeta Function
Lecture 3:   Number Theory III - Newman's Short Proof of the Prime Number Theorem
Lecture 4:   Number Theory IV - Dirichlet Convolution, Dirichlet's Hyperbola Method, and Landau's Theorem on the Mean Value of the Mobius Function
Lecture 5:   Probability - Borel's Law of Large Numbers and Hausdorff's improvement, Random Series and a second proof of Hausdorff's improvement
Lecture 6:   Harmonic Analysis - Maximal Functions and the Lebesgue Differentiation Theorem
Lecture 7:   Ergodic Theory I - An Introduction including Weyl's Equidistribution Theorem and an application to V. I. Arnold's "first digits of the powers of 2" problem
Lecture 8:   Ergodic Theory II - Poincare Recurrence, Ergodicity and von-Neumann's Mean Ergodic Theorem
Lecture 9:   Ergodic Theory III - The Maximal Ergodic Theorem and Birkhoff's Pointwise Ergodic Theorem
Lecture 10: Ergodic Theory IV - Applications of the Pointwise Ergodic Theorem to Normal Numbers and Continued Fractions, specifically Khintchine's constant

Some Notes on Prime Numbers

• Infinitude of Primes
• Prime Number Estimates of Chebyshev (We gave a rather different approach, similar to the argument below)
  
• A simple proof of Bertrand's Postulate
• Relations Equivalent to the Prime Number Theorem
• Arithmetic Functions and Dirichlet Convolution
• Asymptotics for the Mean Value of some Arithmetic Functions

Notes on Ergodic Theory

• Ben Green's Lecture Notes on Ergodic Theory. These conclude with an application to combinatorial number theory, namely Furstenberg's approach to Szemeredi's theorem.