2019/2020
Научно-исследовательский семинар "Аналитическая теория чисел 2"
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Калмынин Александр Борисович
Язык:
английский
Кредиты:
3
Контактные часы:
36
Course Syllabus
Abstract
Analytic number theory is an area of number theory that uses analytic methods to study properties of the integers. No progress towards some famous problems such as Golbach’s conjecture, Waring’s problem or twin primes conjecture would be possible without the development of analytic methods such as bounds for exponential sums and theorems on the distribution of prime numbers. In the first part of the course we will mostly concentrate on the properties of prime numbers, as they are building blocks of integers. We will discuss different proofs of the Prime Number Theorem, distribution of primes in arithmetic progressions and also some basic sieve methods. In the second semester we will learn how to use Fourier-analytic principles (and heuristics) to obtain number-theoretic results. For instance, we will discuss the large sieve method and its numerous applications, such as results on the least quadratic nonresidues, and derive some properties of the Riemann zeta-function from general results on exponential sums and certain bilinear inequalities.
Learning Objectives
- The course is intended to introduce basic concepts and methods of modern analytic number theory to the students and to provide an experience of solving number-theoretic problems.
Expected Learning Outcomes
- At the end of the course, students will be able to find estimates for the exponential sums arising in different areas of analytic number theory and to apply large sieve method to the problems of multiplicative number theory
Course Contents
- The large sieve method and its applications. Generalized Hilbert inequality. Least quadratic nonresidue, sums of unit fractions, Selberg's theorem on primes in very short intervals, equidistribution in residue classes.
- . Equidistribution modulo 1. Discrepancy, Erd\H{o}s-Turan inequality. Equidistribution of values of polynomials. *Van der Corput sets. *Pair correlations.
- Exponential sums. Estimates for Weyl sums. Theory of exponent pairs, A and B processes. Number-theoretic applications. *Vinogradov's method.
- . Properties of the Riemann zeta-function. Hardy-Littlewood approximate functional equation. Mean values and upper bounds for zeta. Results on distribution of zeros.
Bibliography
Recommended Core Bibliography
- Heath-Brown, D. R. (2002). Lectures on sieves. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0209360
Recommended Additional Bibliography
- Ivic, A., Krätzel, E., Kühleitner, M., & Nowak, W. G. (2004). Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0410522