2020/2021
Research Seminar "Character Sums"
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type:
Optional course (faculty)
Where:
Faculty of Mathematics
When:
3, 4 module
Instructors:
Alexander B. Kalmynin
Language:
English
ECTS credits:
3
Contact hours:
36
Course Syllabus
Abstract
Many questions in number theory lead to study of sums of multiplicative functions on the set of natural numbers. One of the simplest and at the same time most insightful examples is the case of periodic multiplicative functions, i.e. Dirichlet characters. Course "Character sums" will be devoted to the study of estimates for the sums of Dirichlet characters over various sets and their applications to other problems in number theory. We will obtain estimates for sums of Dirichlet characters via exponential sums and connect the resulting estimates with properties of corresponding L-functions. We will also discuss connections between character sums and distribution of quadratic residues modulo large prime numbers, bounds for the least quadratic nonresidue and the least prime quadratic residue, large sieve method and its applications.
Learning Objectives
- The course is intended to introduce basic concepts and methods in theory of character sums to the students and to provide an experience of solving number-theoretic problems related to presented results.
Expected Learning Outcomes
- At the end of the course, students will be able to find estimates and formulas for character sums arising in different areas of analytic number theory and to apply large sieve method to the problems of multiplicative number theory.
Course Contents
- Dirichlet characters, their Fourier developments and L-functions, law of quadratic reciprocity. Values of Dirichlet L-functions at s=1 and quadratic fields. *Class numbers and elliptic curves.
- Polya-Vinogradov inequality and its applications. Generalized Riemann hypothesis, conditional estimates for character sums. *Lower bounds for character sums.
- Points on curves over finite field, Stepanov method and Burgess' bound. The least quadratic nonresidue.
- Sums over primes and shifted primes, the least prime quadratic residue and the least primitive root. Large sieve method and Linnik's theorem on the least quadratic nonresidue. *Other applications of the large sieve.
