2023/2024
Analytic Number Theory
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Language:
English
ECTS credits:
6
Contact hours:
60
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.