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 1. Sums and integrals: Thms 3 and 4 in L 1. 2. The average order of tau: Thm. 2 in L 2. 3. The average order of phi: Thm. 4 in L 2. 4. Maximal order of tau: Cor. 20 in L 3. 5. The formula relating beta and Gamma functions: Thm. 8 in L 5. 6. Absolute convergence of infinite products: Thm. 23 in L 5. 7. Prove that zeta is nonzero for sigma greater than or equal to 1: Thm. 25 in L 6.
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
Lecture 10, May 3 (updated May 3, 2024) Part III. Probabilistic Number Theory. Densities. Limiting distributions of arithmetic functions
Lecture 11, May 10 (updated May 10, 2024) Normal order
Lecture 12, May 17 (updated May 17, 2024) Distribution of additive functions and mean values of multiplicative functions
Lecture 13, May 24 (updated May 24, 2024) Friable integers. The saddle point method. Integers free of small prime factors

May 2024