Course description
Analytic number theory is basically about number theoretic problems that are studied by means of methods from analysis. This is a very broad area and it is clearly impossible to cover this in a single course.
We will focus on a few topics. In the first part of the course we discuss prime number theory, and we will give a relatively simple proof, due to Newman, of the prime number theorem for arithmetic progressions, which gives an asymptotic expression for the number of primes in a given arithmetic expression up to x, as x tends to infinity.
In the second part, we consider the famous circle method of Hardy and Littlewood, which is a very powerfool tool to prove that certain Diophantine equations are solvable. To illustrate this method, it is shown that every sufficiently large integer is the sum of nine positive cubes.
For further details we refer to the course website, see the link below.
Prerequisites
Analysis: differential and integral calculus of real functions in several variables, convergence of series, (uniform) convergence of sequences and series of functions, basics of complex analysis (comparable with courses Analysis 1,2 and complex function theory given in Leiden);
Algebra: abelian groups (small subset of course Algebra 1 given in Leiden).
A chapter with additional prerequisites, which the students are expected to read themselves, will be posted on the course website (see the link below).
Literature
Lecture notes and homework assignments are issued during the course and will be posted on the course website (see the link below).
Examination
The examination consist of four homework assignments, issued once in three or four weeks and a written or oral exam (depending on the number of students). Both the average of the four homework assignments and the written or oral exam contribute for 50% to the final grade.
Comments
This course will probably not be given in 2017/2018.