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.
Chebyshev and Mertens
The Prime Polynomial Theorem
Squarefree values taken by polynomials
Squarefree values taken by
polynomials - a function field version
An introduction to the
application of the Selberg Sieve: Bounding π(x).
The Selberg sieve II.
Application: Bounding the number of twin primes.
Mean values of multiplicative functions
Primes in arithmetic progressions I: The case of a fixed modulus.
Primes in arithmetic progressions II: variable moduli and the Brun
The Arithmetic form of the Large Sieve and Linnik's theorem on the
The analytic form of the Large Sieve.
An application of the Large Sieve: Counting reducible polynomials.
Artin's primitive root conjecture
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:
There will be periodic homework assignments which are
Wednesday 11-14, Dan David 204
Contact me at: email@example.com, Office : Schreiber 316, tel: 640-7806
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html