Sieve Theory and its applications 0366.4993.01

Dr. Steve Lester and Prof. Zeev Rudnick

Tel Aviv, Spring 2015

The aim of the course is to study sieves and their applications in analytic number theory. After some background in elementary and analytic number theory, we will start with basic ideas of sieve theory, such as the sieve of Eratosthenes, Selberg's upper bound sieve, and the large sieve. Among applications presented will be: squarefree values of polynomials, upper bounds on the number of twin primes, the Brun-Titchmarsh theorem, Artin's primitive root conjecture.

The class will be given in English


The course is aimed at MSc and Ph.D. students. I will assume knowledge of the courses: Introduction to number theory, complex function theory 1, and basic probability theory.

Lecture notes

  1. Arithmetic functions
  2. Chebyshev and Mertens
  3. The Prime Polynomial Theorem
  4. Erdos-Kac
  5. Squarefree values taken by polynomials
  6. Squarefree values taken by polynomials - a function field version
  7. An introduction to the Selberg sieve.
  8. A first application of the Selberg Sieve: Bounding π(x).
  9. The Selberg sieve II.
  10. Application: Bounding the number of twin primes.
  11. Mean values of multiplicative functions
  12. Primes in arithmetic progressions I: The case of a fixed modulus.
  13. Primes in arithmetic progressions II: variable moduli and the Brun Titchmarsh theorem.
  14. The Arithmetic form of the Large Sieve and Linnik's theorem on the least non-residue.
  15. The analytic form of the Large Sieve.
  16. An application of the Large Sieve: Counting reducible polynomials.
  17. Artin's primitive root conjecture


For the classical background material on the Riemann zeta function etc., consult any of the books


There will be periodic homework assignments which are mandatory .


Wednesday 11-14, Dan David 204

Contact me at:, Office : Schreiber 316, tel: 640-7806

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