2019/2020
Научно-исследовательский семинар "Аналитическая теория чисел 1"
Статус:
Дисциплина общефакультетского пула
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Преподаватели:
Калмынин Александр Борисович
Язык:
английский
Кредиты:
3
Контактные часы:
30
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 bounds and asymptotic formulas for the sums of arithmetical functions using both elementary and analytic methods. Students will demonstrate the ability to apply basic theorems on the distribution of primes and sifted sequences in the context of various number-theoretic problems.
Course Contents
- Arithmetical functions, Dirichlet convolution of arithmetical functions. Partial summation method, Dirichlet's hyperbola method. Average orders of τ_k,σ_k,φ and other arithmetical functions. Mӧbius function, von Mangoldt function.
- Sieve methods: Eratosthenes-Legendre sieve, combinatorial sieves. Brun's constant. There exist infinitely many m such that both m and m-2 have at most nine prime factors.
- Prime numbers in arithmetic progressions. Dirichlet characters and Dirichlet L-functions. Siegel-Walfisz theorem. *Siegel zeros and class number problem.
- . The Prime Number Theorem. Contour integration method. Basic properties of Riemann zeta function, zero-free region, explicit formula. *Tauberian proof of PNT. Selberg symmetry formula, elementary proof of PNT. *Banach algebra proof of PNT.
Bibliography
Recommended Core Bibliography
- Granville, A. (2014). What is the best approach to counting primes? Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1406.3754
Recommended Additional Bibliography
- Gonek, S. M. (2004). Three Lectures on the Riemann Zeta-Function. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0401126