Prime Numbers II

First lecture: Monday, 18/02/2012.

Course Description

This course will study some of the methods available for the study of the distribution of primes, with a focus on primes in interesting sequences and on different types of sums over primes. This will include an introduction to sieve methods and to Vinogradov's method of bilinear forms, and a survey of recent results, for instance of the results of Green and Tao on linear equations in primes.

The goal of the course is to obtain a familiarity with the basic ideas of sieve theory, and their applications, as well as the methods involving bilinear form estimates for estimating sums over primes.

The basic outline will be the following:
* Reminders concerning the distribution of primes in arithmetic progressions
* Sieve methods: Selberg sieve, large sieve, combinatorial sieve
* Applications of sieve methods
* Sums over primes: Vinogradov's approach
* The ternary Goldbach problem and other applications
* Survey of more recent results

Exercises and Solutions

Requirements for the certificate ("Testat"): TBA

Please report any mistakes that you spot in the exercises / solutions.

References

• H. Iwaniec and E. Kowalski: "Analytic number theory", AMS 2004.
• J. Friedlander and H. Iwaniec: "Opera de cribro", AMS 2010.
• B. Green and T. Tao: "Linear equations in primes" (Annals; arXiv).