Analytic Number Theory

A.Y. 2019/2020
Overall hours
Learning objectives
The course introduces the student to the Analytic Number Theory by showing the solutions of some of its classical problems.
Expected learning outcomes
Student will be able to operate with some fundamental tools and results in Analytic Number Theory.
Course syllabus and organization

Single session

Lesson period
First semester
Course syllabus
Prime Number Theorem: Riemann zeta function, zero free region, proof via a Tauberian theorem and elementary proof with the method of Bombieri-Wirsing.

Sieves: Selberg's lambda square method. Brun-Titchmarsh theorem. Brun's result about the twin primes. Romanov Theorem.

Sumsets: Schnirelmann's notion of density. Mann's theorem. Existence of an asymptotic basis for sets having positive density. The Schnirelmann's result about the Goldbach problem. New proof of the Romanov's theorem.

Set of integers and linear progressions: van der Waerden's theorem. the conjecture of Erdös-Turàn and the Roth's theorem. Highlights about the Szemerédi's proof of the conjecture. Linear progressions of prime numbers: existence of triplets of primes (via Roths's theorem) and highlights about the solution of the general problem by Green e Tao (via the Szemerédi's theorem).

The Waring problem: qualitative solution of Linnik and Newmann. Highlights about the quantitative aspects (singular series and the Hardy-Littlewood result).
Prerequisites for admission
Courses in Analysis 1/2/3. A good comprehension of basic results in Complex Anaysis is further strongly welcome.
Teaching methods
Lessons from the teacher.
Teaching Resources
-G. Molteni: Notes for the course in analytic number theory, available on the web page… .
-H. Iwaniec, E. Kowalski: Analytic number theory, AMS Colloquium Publications 53, American Mathematical Society, Providence RI, 2004.
-H. L. Montgomery, R. C. Vaughan: Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.
-P. Pollack: Not always buried deep, A second course in elementary number theory, AMS, Providence RI, 2009.
Assessment methods and Criteria
3/4 Homeworks are provided; their solution is necessary to be elegible for the final exam. A solution in team of the homoworks is possible. The final exam is individual (obviously) and consists essentially in a (long) oral discussion about the main techniques and topics which have been introduced in the course.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Molteni Giuseppe
My office: Dipartimento di Matematica, via Saldini 50, first floor