Department of Mathematics

Probabilistic Number Theory

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Lecturer Prof. Emmanuel Kowalski
Lecture Th 13-14, HG D 5.2
Coordinator Benny Löffel
Exercises Mo 15-16, HG D 7.1
Assistants Alexandru Gabriel Sava    

First lecture: Thursday, September 17, 2015.

Lecture notes: here


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: 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 ( will check the submitted solutions. Please contact him if you have any questions regarding the solutions of the 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

Exercise Sheet Submission date Solution
sheet 1
28.09.2015 solution 1
sheet 2 05.10.2015 solution 2
sheet 3 12.10.2015 solution 3
sheet 4 19.10.2015 solution 4
sheet 5 26.10.2015 solution 5
sheet 6 2.11.2015 solution 6
sheet 7 9.11.2015 solution 7
sheet 8 16.11.2015 solution 8
sheet 9 23.11.2015 solution 9
sheet 10 30.11.2015 solution 10
sheet 11 07.12.2015 solution 11
sheet 12 No submission solution 12



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© 2016 Mathematics Department | Imprint | Disclaimer | 18 February 2016