Sieves in Number Theory

October - December 2008.  Dr. Tim Browning and Professor Roger Heath-Brown.

Syllabus

1. Introductory material
[Open problems; Erastothenes-Legendre; Applications].
2. Large sieve inequality
[Ramanujan sums; least non-residue modulo p; random conics without a rational point].
3. Selberg sieve
[fundamental theorem; Brun-Titchmarsh theorem; discussion of Bombieri-Vinogradov; Titchmarsh divisor problem].
4. Small gaps between primes
[Goldston-Pintz-Yildirim].

Reading (among many possible sources)

• A. Cojocaru and R. Murty, An introduction to sieve methods and their applications, LMS Student Texts, CUP, 2006.
• G. Harmon, Prime-Detecting Sieves, Princeton 2007.
• H. Iwaniec and E. Kowalski, Analytic number theory, AMS 2004.

Course Notes:

Sandro Bettin has very kindly typeset the lecture notes for the lecture course:

• Sieves in number theory

Course Assessment

The course will be assessed by problem sheets that will be assigned during the lectures.

• Problem sheet A
• Problem sheet B

Please send your attempts to Tim Browning, School of Mathematics, University of Bristol, Bristol, BS8 1TW by 10th January 2009.