Probabilistic Number Theory Spring 2021

Lecturer
Emmanuel Kowalski
Coordinator
Ilaria Viglino
Lectures
Mondays, 10 to 12; Thursdays, 14 to 16; Online: Zoom link

Summary

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

The lecture will be accompanied by roughly biweekly exercise classes. We will announce the precise dates in the lecture as well as here. You should submit your exercise sheets in PDF form to the coordinator.

exercise sheet due by solutions
Exercise sheet 1 March 1 Solutions 1
Exercise sheet 2 March 18 Solutions 2
Exercise sheet 3 April 1 Solutions 3
Exercise sheet 4 April 22 Solutions 4
Exercise sheet 5 May 10 Solutions 5
Exercise sheet 6 May 24

Content of the Lectures

We summarize here the contents of the lectures with links to the notes.
DayContent
22.2.2021 Introduction; Statement of the Erdös-Kac Theorem, application to the multiplication table problem; integers in arithmetic progressions, statement of Schoenberg's Theorem
Notes
Recording
25.2.2021 Proof of Schoenberg's Theorem; proof of the criterion B.4.4 for convergence in law using approximations.
Notes
Recording
4.3.2021 Statement of the Erdös-Wintner Theorem; motivation of the limit using Kolmogorov's Three Series Theorem; proof of the convergence part of the theorem using the criterion B.4.4.
Notes
Recording
8.3.2021 Statement and proof of the Erdös-Kac Theorem; tools used include the Lévy convergence criterion (to prove cases of the CLT for independent random variables), and the method of moments and its converse.
Notes
Recording
11.3.2021 Comments on the Erdös-Kac Theorem: generalizations to other types of "random integers" (polynomial values, shifted primes, functions of matrices), and comments on convergence without renormalization. Beginning of Chapter III, statement of the Chebychev-Bias.
Notes
Recording
18.3.2021 Statement of the existence of the Rubinstein-Sarnak measure in the Chebychev bias. Discussion of primes in arithmetic progressions; discussion of characters of finite abelian groups. Definition of Dirichetl L-functions and statement of the Euler product.
Notes
Recording
22.3.2021 Proof of the Euler product. Sketch that the analytic behavior of the Dirichlet L-functions close to 1 give the asymptotic distribution for primes in arithmetic progressions. Definition of the von Mangoldt function, logarithmic derivative formula. Statement of the "explicit formula".
Notes
Recording missing
25.3.2021 Motivation for the explicit formula. Definition of the Mellin transform. Statement of the Generalized Riemann Hypothesis modulo q.
Notes
Recording
1.4.2021 Discussion of GRH. Proposition 5.3.1 of the notes; origin of the Chebychev bias in the leading term when comparing the fluctuations of the prime counting function with the sums involving the von Mangoldt function.
Notes
Recording
12.4.2021 Relation of the random variables in the Chebychev bias to Kronecker's Theorem; explanation of this result. Definition of Haar measure, statement of the Weyl Criterion for convergence to uniform measure on a compact abelian group.
Notes
Recording
15.4.2021 Proof of Kronecker's Theorem and of Th. 5.3.3. Discussion of the reason for the logarithmic weight measure in the Chebychev Bias.
Notes
Recording
22.4.2021 Proof of existence of the Rubinstein-Sarnak measure using B.4.4. Statement of the Simplicity Hypothesis, statement of the formula for the R-S measure assuming this. Example of the original Chebychev bias.
Notes
Recording
26.4.2021 End of discussion of the Chebychev bias, statement of tail bounds for the difference in the original case. Beginning of Chapter IV about distribution of values of the Riemann Zeta function; statement of Bagchi's Theorem and of Voronin's Theorem.
Notes
Recording
29.4.2021 Statement of Selberg's Theorem. Motivation for Bagchi's Theorem and explanation of the limiting random function as random Euler product. Outline of the proof. Statement of Step 1 (existence and series representation for the limiting random function).
Notes
Recording
6.5.2021 Proof of Proposition 3.2.9, including discussion of basic analytic properties of Dirichlet series and of the Menshov-Rademacher Theorem (with the example of Fourier series).
Notes
Recording
10.5.2021 Smoothing formula (A.4.3) for representing Dirichlet series outside of the region of absolute convergence. Statement of the approximation theorem for the random Dirichlet series and the Riemann zeta function (Propositions 3.2.11 and 3.2.12), deduction of Bagchi's Theorem assuming it.
Notes
Recording
17.5.2021 End of the proof of Bagchi's Theorem. Some words on Voronin's Theorem. Beginning of the discussion of exponential sums.
Notes
Recording
27.5.2021 Introduction to exponential sums; example of Jacobi sums, a few words on the circle method as a motivating application. Definition of Kloosterman sums and Kloosterman paths, statement of the limit theorem for the Kloosterman paths, and outline of the steps of the proof.
Notes (first few pages missing because of a crash at the beginning; they are visible in the Zoom recording).
Recording
31.5.2021 Comments on the random Fourier series arising in the limit theorem; comparison with Brownian motion and random walks. Heuristic motivation for the shape of the limit using the completion method. Discussion of criteria for convergence in law in the space of continuous functions. Existence of the random Fourier series. Convergence of Fourier coefficients.
Notes.
Recording
3.6.2021 Proof of tightness for Kloosterman paths. Comments on the proof of Katz's Theorem. Discussion of the problem of fixing one of the two parameters.
Notes.
Recording