Analytic Number Theory

A.Y. 2016/2017
6
Max ECTS
42
Overall hours
SSD
MAT/05
Language
English
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

Responsible
Lesson period
Second semester
Course syllabus
Analytic Number Theory

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).
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Molteni Giuseppe
Professor(s)
Reception:
My office: Dipartimento di Matematica, via Saldini 50, first floor