Analytic Number Theory

A.Y. 2021/2022
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
Second semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
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
Basic courses in Analysis (covered in courses named Analysis 1/2/3), plus the basic course in Complex Analysis.
Teaching methods
Lessons from the teacher.
Teaching Resources
G. Molteni: Notes for the course in analytic number theory, available at the home page of the teacher.

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
In order to pass this exam the student has to produce written (and correct, of course) solutions of the Homeworks I will propose during the course, plus a final oral exam about the main topics of 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