ANALYTIC NUMBER THEORY
Mastermath course - Fall 2018 - 8EC
Classes: Thursdays 14:00-16:45 (2-4:45 pm)
September 13-December 13 or 20, 2018 at the VU (Free University)
Exam: Thursday January 10, 2019, 14:00-17:00 (2-5 pm), room HG 05A00
|Teachers and assistants:||
Dr. Jan-Hendrik Evertse (Universiteit Leiden)
Dr. Damaris Schindler (Universiteit Utrecht)
Assistants (responsible for the exercise classes and the grading of the
The first class on September 13 will be given by Peter Koymans, and according to the present plan the subsequent six classes by Jan-Hendrik Evertse and the last seven classes by Damaris Schindler.
This mastermath course is not identical to the one on Analytic number theory given
in Leiden in the fall of 2016, but there is a considerable overlap.
Students who got a grade for the Leiden course cannot do exam for this mastermath
course unless they drop their grade for the Leiden course.
Rules for the written exam:
Material for the written exam:
Damaris' part: Everything that has been discussed during Damaris' classes (have a look at the videos if you missed some of them), and the seven exercise sheets. More precisely, the following topics can be examined: sieving for squre-free values of polynomials; Brun's sieve; the large sieve inequality and its applications to sieving; Gauss sums (the material from Section 3.4 from the lecture notes); the Barban-Davenport-Halberstam theorem.
At the written exam you may expect questions of the following type:
Computation of the grades:
Final grade for analytic number theory:= (4*(Homework grade)+6*(grade for written exam))/10.
Damaris' part has not been taught before so there are no old exam exercises about that.
Analysis: differential and integral calculus
of real functions in several variables, convergence of series,
(uniform) convergence of sequences of functions,
basics of complex analysis;
Algebra: elementary group theory, mostly only about abelian groups.
Chapter 0 of the course notes (see below) gives an overview of what will be used during the course. We will not discuss the contents of Chapter 0 and they will not be examined, but you are supposed to be familiar with the theorems mentioned in Chapter 0.
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,
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, including the fundamental lemma of sieve theory, Selberg's method, bilinear form methods, and the Large Sieve inequality. If time permits, we will discuss in more depth primes in arithmetic progressions and the Bombieri-Vinogradov Theorem, as well as applications to the ternary Goldbach problem (representation of odd integers as a sum of three primes).
Homework assignments and their deadlines of delivery will be posted here.
Please note that these deadlines are strict. Assignments are posted about two
weeks prior to the deadline.
Assignments: (exercises are taken from the lecture notes; in parentheses the maximal number of points for an exercise)
|Course notes (pdf):
In Chapter 0 we have collected some facts
from algebra and analysis that will be used in the course.
The contents of Chapter 0 will not be discussed in the course and they will not be examined, but all theorems, corollaries etc.
are assumed to be known.
Chapter 0: Notation and prerequisites
This completes the notes of Jan-Hendrik Evertse's part of the course.
Damaris Schindler will lecture from the books "Opera de Cribro" (OdC) by Friedlander and Iwaniec,
and "Analytic Number Theory" (ANT) by Iwaniec and Kowalski.
Below you find an overview of the chapters that are taught, together with the sheets with the exercises for the exercise classes.
This course will probably not be given in 2019/2020.
This course is recommended for a Master's thesis project in Number Theory.
Recommended for further reading:
This book discusses the properties of the Riemann zeta function, as well as those of Dirichlet L-functions. Further it gives the proofs of de la Vallée Poussin of the prime number theorem and the prime number theorem for arithmetic progressions. Lastly, it treats some sieve theory.
In this book, Davenport considers various classes of Diophantine equations and inequalities to be solved in integers. Under certain hypotheses he shows that these are solvable and obtains asymptotic formulas for the number of solutions whose coordinates have absolute values at most X, as X→∞. A particular instance of this is Waring's problem, that for every positive integer k≥3 there is g such that every positive integer can be expressed as a sum of non-negative k-th powers.The proofs are based on the circle method of Hardy and Littlewood.
Sieve theory and applications to primes in arithmetic progressions.
On-line survey paper on estimates for π(x) (number of primes up to x) and related issues.
Classic book on the distribution of prime numbers.
Analytic number theory bible, containing a lot of material. The proofs are rather sketchy.
This book gives both a proof of the Prime Number Theorem based on complex analysis which is similar to the one we give during the course, as well as an elementary proof not using complex analysis. The book should be accessible to third year students.
The third part contains analytic number theory related to algebraic number theory, such as a proof of the functional equation of the Dedekind zeta function for algebraic number fields (this is a generalization of the Riemann zeta function), a proof of the functional equation for L-series related to Hecke characters (generalizations of Dirichlet characters), a proof of the Prime Ideal Theorem (a generalization of the Prime Number Theorem). We will not discuss these topics during our course, but it is important related material.
This book gives a comprehensive introduction to complex analysis. It includes topics relevant for number theory, such as elliptic functions and a simple proof of the Prime Number Theorem, due to Newman.
This book discusses in detail the Riemann zeta function, L-functions, the Prime Number Theorem for arithmetic progressions and refinements thereof, and a brief introduction to sieve theory.
This book gives an introduction to analytic number theory, including a simple proof of the Prime Number Theorem, and various other topics, such as an asymptotic formula for the number of partitions, Waring's problem about the representation of integers by sums of k-th powers, etc.
The title speaks for itself.
This book discusses several applications of the Hardy-Littlewood circle method, such as Diophantine equations and inequalities, Waring's problem (like Davenport's book above, but with more recent refinements) and the ternary Goldbach problem (that every odd integer larger than 5 is the sum of three primes).
It may also be interesting to have a look at Riemann's memoir from 1859 (you can download both the original German version and its English translation), in which he proved several results on the function ζ(s)=∑n≥1 n-s and among other things formulated the Riemann Hypothesis (in a different but equivalent form).
Official classification of mathematics subjects. Number theory is classified under no. 11.
Website for the number theory community with many useful links.
Long list of downloadable lecture notes on various branches of number theory including analytic number theory.
Online mathematical data bases which can be used to find abstracts of mathematical papers, lists of papers of mathematicians, etc. MathSciNet covers the period 1940-... and Zentralblatt 1930-... . These websites are accessible only to subscribers. If your department has subscribed to them, you may consult them through your department's network.
An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.
Freely accessible mathematical preprints archive; number theory preprints are categorized under NT.
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