The class will be given in English

Prerequisites

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

Bibliography

• A.C. Cojocaru and M.R. Murty, An Introduction to Sieve Methods and Their Applications Cambridge University Press, 2006
• H. Halberstam and H.E. Richert, Sieve Methods. Courier Dover Publications, 2013.

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

• H. Davenport, Multiplicative Number Theory
• T. Apostol, Introduction to Analytic Number Theory.
• Hugh L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I: Classical Theory

Homework

• Assignment 1, Due date March 30
• Assignment 2, Due date April 16, 2015.
• Assignment 3, Due date May 27, 2015.
• Assignment 4, Due date June 17, 2015.
• Takehome Exam, to be returned to me be email or to my mailbox by July 15, 2015.

Schedule

Course homepage: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html