OVERVIEW

The course will give an introduction to analytic number theroy.

The lecture will be in Italian unless requested otherwise by the students.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce basic elementary and analytical concepts, and the relative techniques, for the study of arithmetic problems, in particular concerning prime numbers. The course provides analytical prerequisites necessary to address more advanced issues in Number Theory, Arithmetic Geometry and related topics.

AIMS AND LEARNING OUTCOMES

The aim of the course is to introduce basic elementary and analytical concepts, and the relative techniques, for the study of arithmetic problems, in particular concerning prime numbers. The course provides analytical prerequisites necessary to address more advanced issues in Number Theory, Arithmetic Geometry and related topics. It is expected that by the end of the course the students have mastered the basic techniques of analytic number theory and that they can employ them to solve arithmetic problems.

TEACHING METHODS

Traditional

SYLLABUS/CONTENT

Arithmetical functions: arithmetical and algebraic aspects, asymptotic behavior. Elementary methods for the distribution of primes: Euler, Legendre and Chebyshev. Elements of cryptography. Complements of Analysis: Dirichlet series, Mellin transform and Poisson formula. Riemann zeta function: general properties and distribution of zeros. Prime Number Theorem: explicit formulae and PNT with remainder. Dirichlet L-functions: Dirichlet characters, general properties of L-functions and distribution of zeros. Dirichlet's theorem.

RECOMMENDED READING/BIBLIOGRAPHY

rse notes.

A.E.Ingham - The Distribution of Prime Numbers - Cambridege U.P. 1964.

H.Davenport - Multiplicative Number Theory - Springer 1980.

G.Tenenbaum, M.Mendes-France - The Prime Numbers and Their Distribution - AMS 2000.

G.Tenenbaum - Introduction to Analytic and Probabilistic Number Theory - Cambridge U.P., 1995

TEACHERS AND EXAM BOARD

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and Oral exam. Students with a grade greater than or equal to 14 are admitted to the oral exam. The final grade is based on the grade of the written exam, but can be increased or decreased depending on the student's performance in the oral exam.

ASSESSMENT METHODS

Evaluation of written and oral examination. In the written part, some exercises will be proposed, and the quality of the solutions written by the students will be evaluated. The oral part deals mainly with the theory developed during the course, and the understanding of the theorems and the ability of reproducing proofs of the students will be evaluated.

FURTHER INFORMATION

Students with DSA should follow the instructions given in Aulaweb: https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3