Analytic and Combinatorial Number Theory, NDMI045, summer term 2023/24
My plan is to survey the
textbook
Introduction to Analytic and Probabilistic Number Theory (Third Edition) by G. Tenenbaum. I will survey every result (labeled as a Theorem,
Proposition, Lemma and Defition) in the book,
and prove a sample of them. I will produce lecture notes to my lectures, see below.
Exam questions to be updated
Lecture 1, February 23 (updated February 21, 2024) Part I. Elementary Methods.
Some tools from real analysis. Prime numbers. Arithmetic functions
Lecture 2, March 1 (updated February 29, 2024) Average orders. Brun's sieve and
Linnik's large sieve
Lecture 3, March 8 (updated March 11, 2024 - the mention of Landau's asymptotics
of the number of two-square numbers added) Selberg's sieve. Extremal orders
Lecture 4, March 15 (updated March 15, 2024) The method of van der Corput. Diophantine approximation
Lecture 5, March 22 (updated March 22, 2024) Part II. Complex Analysis Methods.
The function Gamma. Dirichlet series
March 29 - Good Friday, no lecture
Lecture 6, April 5 (updated April 5, 2024) Summation formulae. The Riemann zeta function
Lecture 7, April 12 (updated April 12, 2024) The prime number theorem and the Riemann
hypothesis. The Selberg--Delange method. Two arithmetic applications
Lecture 8, April 19 (updated April 19, 2024) Tauberian Theorems
Lecture 9, April 29 (updated April 26, 2024) Primes in arithmetic progressions
April 2024