Probabilistic Number Theory

First lecture: Thursday, September 17, 2015.

Lecture notes: here

Contents

The goal of the course is to present some results of probabilistic number theory in a unified manner. The main concepts will be presented in parallel with the proof of three main theorems:
(1) the Erdös-Kac theorem and its variants concerning the number of prime divisors of integers in various sequences;
(2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line;
(3) functional limit theorems for the paths of partial sums of families of exponential sums such as Kloosterman sums.

Prerequisites

Prerequisites: Complex analysis, measure and integral; some probability theory is useful but the main concepts needed will be recalled. Some knowledge of number theory is useful but the main results will be summarized.

Teaching Assistants

The exercise sessions will be given by Benny Löffel.

Alexandru Gabriel Sava (alexandru.sava@math.ethz.ch) will check the submitted solutions. Please contact him if you have any questions regarding the solutions of the exercises.

Exercises

For students who plan to attend the exam, we strongly recommend to solve the exercises and to hand in their solutions.

First exercise session: 21.09.2015

Literature

• H. Iwaniec and E. Kowalski: "Analytic number theory"