Mastermath course -  Fall 2018 - 8EC

[ Schedule Mastermath Fall 2018  |  Mastermath webpage  |  Homework assignments  |  Course notes ]

Time/Place: Classes: Thursdays   14:00-16:45 (2-4:45 pm)   September 13-December 13 or 20, 2018 at the VU (Free University) (see map)

Class rooms:
Sep 13,20,27, Oct 4,11,18WN P647 (W&N-building, De Boelelaan 1081, floor no. 6, dept. P)
Oct 25HG 02A24 (Main building (hoofdgebouw), De Boelelaan 1105, floor no. 2)
Nov 1,8,15,22,29, Dec 6,13WN P647
Dec 20HG 02A24

Exam: Thursday January 10, 2019,   14:00-17:00 (2-5 pm),  room HG 05A00
Retake: Thursday February 14, 2019,   14:00-17:00,  room WN S631

Teachers and assistants: Teachers:

Dr. Jan-Hendrik Evertse (Universiteit Leiden)
Snellius,   Niels Bohrweg 1,  2333 CA Leiden,  office 248
tel. 071-5277148,   email evertse at

Dr. Damaris Schindler (Universiteit Utrecht)
Hans Freudenthalgebouw,  Budapestlaan 6,  3584 CD Utrecht,  office 615
tel. 030-2539221,   email d.schindler at

Assistants (responsible for the exercise classes and the grading of the homework):
Peter Koymans (Universiteit Leiden),  email peter.koymans at
Lasse Grimmelt (Universiteit Utrecht),  email l.p.grimmelt at

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.

Examination: 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:
Students are not allowed to use books, lecture notes, notebooks, smartphones, or any device with a memory capacity.
Students have to write down their names, university and student numbers on each sheet of exam paper; these should all be very well readable, and names should be CAPITALIZED.
Students will not get points for an exercise if they have given an answer without explanation (even if the answer is correct). It is best to give always an explanation, even if you can not complete it. Then you may not get the maximal number of points for an exercise but at least part of it.
On the exam you are allowed to use without proof all theorems from the lecture notes or those that have been discussed during the classes, unless you are explicity asked to give a proof. You are not allowed to use without proof the results from the exercises. So if you want to use such a result in your solution you have to produce a correct proof.

Material for the written exam:
We may ask questions about the following:
Jan-Hendrik's part:
Chapters 1-6 of the lecture notes, both theory and exercises, with the following exceptions: Sections 3.5, 3.6 and exercises 3.5,3.6,3.7; Sections 4.4, 4.5. We may ask you to apply the functional equations of the zeta-function or L-functions, but you don't have to study their proofs. Further, you do not have to know the proofs of the Tauberian theorems but you have to know them and be able to apply them.

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:
- prove a theorem or result from the lecture notes or from Damaris' classes (as long as the proof is shorter than one page, say; of course we will not ask you to reproduce long proofs);
- applications of the theorems from the course;
- solving an exercise such as in the lecture notes or the exercise sheets or a variation thereof.

Computation of the grades:
Since the amount of homework in the first three assignments given by Jan-Hendrik was somewhat larger than the amount of the last four assignments given by Damaris, the first three assignments will get a heavier weight. The homework grade will be computed as follows:
A:= average of assignments 1,2,3;
B:= average of assignments 4,5,6,7;
Homework grade := (3A+2B)/5.

Final grade for analytic number theory:= (4*(Homework grade)+6*(grade for written exam))/10.

Old exams:
Exam 2014-2015 (covers only the material of Jan-Hendrik's part)
Exam 2016-2017; (covers Jan-Hendrik's part and Efthymios Sofos' part about the circle method)

Damaris' part has not been taught before so there are no old exam exercises about that.

Prerequisites: 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.
Course description: 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, 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: 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.

Practical matters:

  • Do not forget to write (very well readable) or type your name, university and student number on your homework.
  • You may either submit your homework at the course, or electronically by email to both assistants.
  • To simplify grading we prefer very much that you submit the pdf of a (la)tex-file. Handwritten homework will be graded only if it is very well readable and has no erasures. Scanned handwritten homework submitted by email will be accepted only if it is by means of a single pdf-file. Photographs of individual pages will not be accepted.

Assignments: (exercises are taken from the lecture notes; in parentheses the maximal number of points for an exercise)

  • Assignment 1, Due October 4
    1.4 (a:10; b:5; c:5; d:5); 1.9 (a:5; b:5; c:10); 2.5 (10); 2.6 (5); 2.8 (a:5; b:5; c:10); grade =(number of points)/8

  • Assignment 2, Due October 18
    2.7 (a:5, b:10 (you don't have to do limsupn→∞ τ(n)loglog n/log n =log 2)); 3.3 (10); 4.2 (a:5, b:5); 4.5 (a:5, b: 5, c:5); grade =(number of points)/5

  • Assignment 3, Due November 8
    1.3 a,b (not c) (a:5, b:5; this exercise is relevant for Chapter 6); 5.4 (a:5, b:5, c:5; you have to use Exercise 5.3 and the theorems and corollaries mentioned in Exercise 5.2); 6.3 a,b (not c,d) (a:5, b:5; recall -ζ'(s)/ζ(s)=∑n≥1 Λ(n)n-s, ψ(n)=∑n≤xΛ(n)); 6.5 (a:5, b:5, c:5; use that L(s,χ)-1=∑n≥1 μ(n)χ(n)n-s); grade =(number of points)/5

  • Assignment 4, Due November 22
    Exercises 2 and 3 of the first exercise sheet.

  • Assignment 5, Due November 29
    Exercise 2 of the second exercise sheet.

  • Assignment 6, Due December 6
    Exercise 4 of the third exercise sheet (exercise 4 has been modified on November 23).

  • Assignment 7, Due December 13
    Exercise 1 of the fourth exercise sheet.
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
Chapter 1: Introduction to prime number theory  (exercise 1.8 corrected on 28/12)
Chapter 2: Dirichlet series and arithmetic functions
Chapter 3: Characters and Gauss sums
Chapter 4: The Riemann zeta function and L-functions (with correction in exercise 4.4c)
Chapter 5: Tauberian theorems
Chapter 6: The Prime Number Theorem for arithmetic progressions

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.
November 8: OdC, Chapter 1  1st exercise sheet
November 15: OdC, Chapter 6.1,6.2  2nd exercise sheet
November 22: ANT, Chapters 7.3,7.4 (section before 7.5) and 7.4 (section after 7.5)   3rd exercise sheet (modified on November 23)
November 29: ANT, Chapters 7.3,7.4,7.4   4th exercise sheet
December 6: ANT, Chapters 7.3,7.4,7.4   5th exercise sheet
December 13: Section 3.4 of Jan-Hendrik Evertse's lecture notes; ANT Chapters 7.5, 7.6, 17   6th exercise sheet
December 20: ANT Chapters 7.5, 7.6, 17  7th exercise sheet

Remarks: This course will probably not be given in 2019/2020.
This course is recommended for a Master's thesis project in Number Theory.
Literature: Recommended for further reading:
  • H. Davenport, Multiplicative Number Theory (2nd edition), Springer Verlag, Graduate Texts in Mathematics 74, 1980
    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.
    ISBN 3-540-90533-2

  • H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, Cambridge University Press, 1963, reissued in 2005 in the Cambridge Mathematical Library series.
    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.
    ISBN 0-521-60583-0

  • J. Friedlander, H. Iwaniec, Opera de Cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society 2010.
    Sieve theory and applications to primes in arithmetic progressions.

  • A. Granville, What is the best approach to counting primes, arXiv:1406.3754 [math.NT].
    On-line survey paper on estimates for π(x) (number of primes up to x) and related issues.

  • A.E. Ingham, The distribution of prime numbers, Cambridge University Press, 1932, reissued in 1990
    Classic book on the distribution of prime numbers.
    ISBN 0-521-39789-8

  • H. Iwaniec, E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society 2004.
    Analytic number theory bible, containing a lot of material. The proofs are rather sketchy.
    ISBN 0-8218-3633-1

  • G.J.O. Jameson, The Prime Number Theorem, London Mathematical Society Student Texts 53, Cambridge University Press 2003.
    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.
    ISBN 0-521-89110-8

  • S. Lang, Algebraic Number Theory, Addison-Wesley, 1970.
    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.
    ISBN 0-201-04201-0

  • S. Lang, Complex Analysis (4th edition), Springer Verlag, Graduate Texts in Mathematics 103, 1999
    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.
    ISBN 0-387-98592-1

  • H.L. Montgomery, R.C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, Cambridge studies in advanced mathematics 97, 2007
    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.
    ISBN-10 0-521-84903-9

  • D.J. Newman, Analytic Number Theory, Springer Verlag, Graduate Texts in Mathematics 177, 1998.
    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.
    ISBN 0-387-98308-2

  • E.C. Titchmarsh, The theory of the Riemann zeta function (2nd edition), revised by D.R. Heath-Brown), Oxford Science Publications, Clarendon Press Oxford, 1986.
    The title speaks for itself.
    ISBN 0-19-853369-1

  • R.C. Vaughan, The Hardy-Littlewood method (2nd edition, Cambridge University Press, 1997.
    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).
    ISBN 0-521-57347-5
  • 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).

  • 2010 Mathematics Subject Classification (MSC2010)
    Official classification of mathematics subjects. Number theory is classified under no. 11.

  • Number Theory Web
    Website for the number theory community with many useful links.

  • Online number theory lecture notes
    Long list of downloadable lecture notes on various branches of number theory including analytic number theory.

  • MathSciNet,   Zentralblatt
    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.

  • MacTutor History of Mathematics archive
    An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.

  • Math arXiv
    Freely accessible mathematical preprints archive; number theory preprints are categorized under NT.
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