Course info

Prerequisites:

Elementary algebra (group theory); elementary analysis (real functions in one and more variables).

During the course we will use complex analysis. It would be helpful if you have followed a course on complex analysis so that you will have some basic comfort with the subject, but it is not strictly necessary. Prior to the start of our course, we will post a chapter containing all of the prerequisite material from complex analysis that will be used.


Aim of the course:

We would like to make the students familiar with the basics of prime number theory (first part taught by Jan-Hendrik Evertse) and sieve theory (second part taught by Lola Thompson).


Detailed description of the contents of the course:

The first part of the course is about prime number theory. Our ultimate goal is to prove the prime number theorem, and more generally, the prime number theorem for arithmetic progressions. The prime number theorem, proved by Hadamard and de la Valleé Poussin in 1896, asserts that if π(x) denotes the number of primes up to x, then π(x)∼x/log x as x→∞, that is, limx→∞π(x)(log x/x)=1. The prime number theory for arithmetic progressions, proved by de la Valleé Poussin in 1899, can be stated as follows. Let a,q be integers such that 0<a<q and a is coprime with q and let π(x;a,q) denote the number of primes not exceeding x in the arithmetic progression a,a+q,a+2q,a+3q,... . Let φ(q) denote the number of positive integers smaller than q that are coprime with q. Then π(x;a,q)∼φ(q)-1x/log x as x→∞. This shows that roughly speaking, the primes are evenly distributed over the prime residue classes modulo q.


In the course we will start with elementary prime number theory and then discuss the necessary ingredients to prove the above results: Dirichlet series, Dirichlet characters, the Riemann zeta function and L-functions and properties thereof, in particular that the (analytic continuations of) the L-functions do not vanish on the line of complex numbers with real part equal to 1. We then prove the prime number theorem for arithmetic progressions by means of a relatively simple method based on complex analysis, developed by Newman around 1980.


In the second part, we will give an introduction to sieve methods. Sieves are used to bound the size of a set after elements with certain ``undesirable’’ properties have been removed. A basic example is the method of inclusion-exclusion, which gives an exact count for the number of elements in a set. Most sieves are not as exact, nor as user-friendly, as inclusion-exclusion. However, they are powerful tools for giving (approximate) answers to the question ``How many numbers are there with a given property?’’ 


Sieves have been used for thousands of years, dating back to Eratosthenes. The Sieve of Eratosthenes is used to generate a table of prime numbers by systematically removing all integers with ``small’’ primes as proper divisors. In modern times, more-sophisticated sieves have been developed (by Brun, Selberg, Linnik, and others) to attack famous unsolved problems in number theory, such as the Twin Primes Conjecture and the Goldbach Conjecture. While these problems are still unsolved, we will see how sieves can shed some light on them.  

Homework/exam:

There will be approximately 8 homework assignments and a written exam. Students will have an assignment due every other week. The assignment with the lowest score will be dropped at the end of the semester. The assignments together will contribute 25% and the written exam will contribute 75% to the final grade. The grade for the written exam must be at least 5.



The retake will, in principle, be in written form as well, unless the number of students that want to do it is so small that an oral retake is more efficient. The homework still counts 25%.


Lecture notes/Literature: PDFs with the lecture notes will be posted on the course webpage.


Lecturers: Jan-Hendrik Evertse (UL) (first part), Lola Thompson (UU) (second part)