# Probabilistic Number Theory Spring 2020

Lecturer
Emmanuel Kowalski
Coordinator
Maxim Gerspach

## 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.

## Exercises

New exercise sheets will be posted here at an appropriate time. You can hand in your exercise sheets via email to maxim dot gerspach at math dot ethz dot ch.

exercise sheet due by solutions
Sheet 01 Wed 04.03. Solution 01
Sheet 02 Wed 11.03.

timeroom
Mo 10-12HG D 3.2
Th 13-15HG D 5.2

## Content of the Lectures

To help students who cannot attend some of the lectures, we describe their contents in some detail, and add links to the corresponding handwritten notes.
DayContent
5.3.2020 Proof of the Erdös-Kac Theorem in the form of Theorem 2.3.1. This uses the method of moments, which is summarized in the case of real-valued random variables, referring to Prop. B.5.5 and B.5.6. Example B.5.7 gives a slightly more general form of the statement than was used to finish the computation of the moments. Discussion of the arithmetic ingredients of the proof, namely the Mertens formula for the sum of inverses of the primes up to a bound, and the basic Theorem 1.3.1 concerning integers in arithmetic progressions. (Not yet added to the script.)
Notes
9.3.2020 Proof of the special case of the Central Limit Theorem that is used to conclude the proof of the Erdös-Kac Theorem (Theorem B.7.1 of the lecture notes); this uses the definition of the characteristic function of a random variable with values in a finite-dimensional real vector space, and the Lévy Criterion for convergence in law (beginning of Section B.5 up to Theorem B.5.1). End of the discussion of the Erdös-Kac Theorem with remarks on other proofs from Section 2.4 (in particular with the definition of Poisson random variables.) Begin the next Chapter concerning the Chebychev Bias (Chapter 5 of the lecture notes); notation for primes in arithmetic progressions and statement of the basic theorem of Dirichlet, Hadamard and de la Vallée Poussin concerning the distribution of primes in arithmetic progressions (Theorem C.3.7).
12.3.2020 (1) Definition of the probability spaces Omega_X and the arithmetic random variables N_X that encapsulate the properly scaled sizes of the error terms in the Dirichlet-Hadamard-de la Vallée Poussin theorem on primes in arithmetic progressions (beginning of Section 5.2) (2) Statement of Theorem 1 concerning the Chebychev bias, namely the existence of the Rubinstein-Sarnak distribution N_q (Theorem 5.2.2 in the notes). (3) Remarks, including the statement of the equivalent form of GRH in terms of the size of the error term; reminder of the definition of the support of a random variable (B.2 in the notes); explanation of why one can expect that the support of N_q is the in hyperplane indicated in Th. 5.2.2 (in the script, this is the computation that is done just at the end of Section 5.3). (4) Beginning of outline of the proof: definition of the characters of an abelian group (in the script, this is in Section B.6, after the discussion of Haar measure); example of finite abelian groups (Example B.6.2 (3)), with a sketch of proof. (5) Using the orthogonality of characters, expression of pi(x;q,a) in terms of sums of the values of characters over primes. (This is the beginning of Lemma 5.3.1).