2020/2021
Research Seminar "Distribution of Prime Numbers"
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Where:
Faculty of Mathematics
When:
1, 2 module
Instructors:
Alexander B. Kalmynin
Language:
English
ECTS credits:
3
Contact hours:
30
Course Syllabus
Abstract
The set of prime numbers and its properties, such as distribution of primes in short intervals and arithmetic progressions, plays crucial role in modern number theory, as prime numbers are «bulding blocks» of integers. Course «Distribution of prime numbers» will be devoted to various results about prime numbers, starting from the most classical, such as the Prime Number Theorem and its connection with the Riemann Hypothesis, and ending with more modern and technically difficult ones, such as Vinogradov’s three primes theorem and Heath–Brown’s result on infinitude of twin primes in the case of existence of Siegel zeros. We will also touch on the questions on primes in short intervals and polynomial sequences and learn basics of sieve methods and other methods of analytic number theory along the way.
Learning Objectives
- The course is intended to present basic and modern results in theory of prime numbers 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 asymptotic formulas for different counting functions of prime numbers arising in various areas of analytic number theory and to apply simplest sieve methods.
Course Contents
- Summation of arithmetical functions, Dirichlet series, Perron's formula
- Riemann zeta function, Dirichlet L-functions, Prime Number Theorem, primes in arithmetic progressions, Riemann hypothesis, Siegel-Walfisz theorem, Siegel zeros.
- Sieve methods: sieve of Eratosthenes, Brun's theorem, Vinogradov's formula for von Mangoldt function and estimate for linear exponential sum with primes.
- Any large enough natural number is a sum of three primes. Numbers of the form n^2-1 and n^2+1 with few prime factors. Romanov's theorem.
- Smooth numbers, primes in short intervals and Rankin's theorem on large gaps between primes. *Irregularities in distribution of primes. *Twin primes and Siegel zeros.
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
- 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