Math
8910
 Seminar in Analysis  Summer 2018
Analysis
and Number Theory
MondayFriday
10:0011:00, June 1822 & 2529, in Boyd 410
10 Lectures on Analysis and Number Theory
Lecture
1: Number Theory I  Basic Prime
Number Theory
Lecture 2: Number Theory II  The
RiemannZeta 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
vonNeumann'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
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.
