Meeting days: Tuesday, Friday, 12:10PM to 12:30PM, Hill 525
Syllabus: The course will roughly follow the genesis of probabilistic ideas in analytic number theory.
We will start with probabilistic number theory, a subject which studies the distribution of additive and multiplicative functions. The most basic question here is, for example: What is the number of prime factors of a typical integer?
We will cover classical results such as the Turan-Kublius inequality, the Erdos-Kac theorem on the normality of the distribution of the number of prime factors of n, and the Kubilius model. Time permitting, we will discuss powerful analytic methods, such as the Selberg-Delange method and Halasz's theorem, which are the backbone of the recent theory of "pretentious multiplicative functions" developed by Granville and Soundararajan.
In the second part of the course, we will study the value distribution of zeta and L-functions, drawing analogies with the first part. A typical question here is: What is the average size of the Riemann zeta-function, or of an L-function at the central point? The later case of L-functions at the central points has applications, for example to elliptic curves.
We will discuss Bohr's work on the value distribution of the zeta-function to the right of the crtical line, Selberg's central limit theorem for the critical line and recent work on moments of L-functions which draws heavily from probabilistic techniques. We will address briefly the random matrix theory aspect of those questions.
Time permitting, we will also cover applications of number theory to probability theory.
Grading: No exams or midterms. Undergraduate students are required to hand in before May 7th, at least half of the exercices which will appear in the course notes throughout the semester.
Course notes : Will be updated throughout the semester roughly at two weeks intervals.