A DMATH forum is available for discussion and to ask/answer questions about the course, see here.
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 |
Day | Content |
---|---|
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 |