Analytic Number Theory
A.Y. 2023/2024
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second 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).
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), and a 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.
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
Educational website(s)
Professor(s)
Reception:
My office: Dipartimento di Matematica, via Saldini 50, first floor, Room 1044.