ANALYTIC NUMBER THEORY
Mastermath course - Fall 2022 - 8EC
Classes: Thursdays September 15-December 15, 14:00-16:45 (2-4:45 pm)
Usually, from 14:00-15:45 there will be a lecture, and from 16:00-16:45 an exercise class, where you can try some exercises under guidance of the assistants.
Classes will be given at the VU, Amsterdam, lecture room NU 3A-65 (New University Building, De Boelelaan 1111).
Exam: Thursday January 26, 2023, 13:45-16:45 room NU 2C-33.
Retake: Thursday February 23, 2023, 14:00-17:00 lecture room will be announced.
|Teachers and assistants:||
Dr. Jan-Hendrik Evertse (Universiteit Leiden) email: evertse at math.leidenuniv.nl
Dr. Lola Thompson (Universiteit Utrecht) email: L.Thompson at uu.nl
Assistants (responsible for the exercise classes and the grading of the
If you have a (mathematical or non-mathematical) question that may be of interest to the other students as well you can post this on the mastermath zulip server, under the analytic number theory stream.
The examination will consist of twelve weekly homework assignments and a written exam.
Exam 2014-2015 (covers only the material on prime number theory)
Exam 2016-2017 (material covered by this exam partly differs from what we will do this year)
Exam 2020-2021 (material covered by this exam partly differs from what we will do this year)
Analysis: differential and integral calculus
of real functions in several variables, convergence of series,
(uniform) convergence of sequences of functions.
It will be useful to have some knowledge of the basics of complex analysis. Everything which is needed from complex analysis is contained in Chapter 0.
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 and concepts discussed in Chapter 0.
The first part of the course, taught by Jan-Hendrik, is about prime number theory.
The 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 theorem 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.
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.
The second part, taught by Lola, is 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.
|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, concepts etc. are assumed to be known. You may use this as a reference source during the course.
Chapter 0: Notation and prerequisites
This is all. The interested reader may consult Andrew Granville's paper Primes in intervals of bounded length on the recent work of Zhang and Maynard.
This course will probably not be given in 2023/2024.
This course is recommended for a Master's thesis project in Number Theory.
Recommended for further reading:
This is a very readable book on sieve methods.
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 g non-negative k-th powers.The proofs are based on the circle method of Hardy and Littlewood.
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.
This book focuses on elementary proofs of results from analytic number theory (largely avoiding complex analysis).
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-... . MathSciNet is accessible only to subscribers. You may consult it through your department's network if your department has subscribed to it. Zentralblatt is now open access.
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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