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 BrunTitchmarsh theorem, Artin's primitive root conjecture.
The class will be given in English
Prerequisites
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

Arithmetic functions

Chebyshev and Mertens

The Prime Polynomial Theorem

ErdosKac

Squarefree values taken by polynomials

Squarefree values taken by
polynomials  a function field version

An introduction to the
Selberg sieve.

A first
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
Titchmarsh theorem.

The Arithmetic form of the Large Sieve and Linnik's theorem on the
least nonresidue.

The analytic form of the Large Sieve.

An application of the Large Sieve: Counting reducible polynomials.

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
There will be periodic homework assignments which are
mandatory .
Schedule
Wednesday 1114, Dan David 204
Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 6407806
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html