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401-3109-65L 8 Credits BSC , DR , MSC D-MATH

Probabilistic Number Theory

Lecturers & Examiners: Prof. Dr. Emmanuel Kowalski

VVZ CR n/a

Last Updated: 2026-02-05 15:54:12

Abstract

The course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.

Objective

The goal of the course is to present some results of probabilistic number theory in a unified manner.

Content

The main concepts will be presented in parallel with the proof of a few main theorems: (1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions; (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) the Chebychev bias for primes in arithmetic progressions; (4) functional limit theorems for the paths of partial sums of families of exponential sums.

Resources

Lecture Notes

The lecture notes for the class are available athttps://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf

Learning Materials (Links)

Main link
Lecture Homepage

General Information

Language: English
Levels: BSC , DR , MSC

Examination

Type: session examination
Mode: oral 20 minutes

Course Components

Type	Title	Time & Place	Hours
lecture with exercise	Probabilistic Number Theory	Mon 10:15-12:00 (ML F 39) Thu 14:15-16:00 (HG D 5.2)	4 h weekly

Offered In

Mathematics Bachelor
- Electives
  - Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
Mathematics Master
- Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
  - Electives: Pure Mathematics
    - Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
Doctoral Department of Mathematics (More Information at: The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. WARNING: Do not mistake ECTS credits for credit points for doctoral studies!)
- Graduate School (Official website of the Zurich Graduate School in Mathematics:)