Course for 3rd year bachelor and master students -  Fall 2016

[e-studiegids Bachelor wiskunde  |  e-prospectus Master mathematics  |  Homepage Mathematisch Instituut ]
Seal of university
Universiteit Leiden
Mathematisch Instituut

Teachers: Dr. Jan-Hendrik Evertse
office 248,  tel. 071-5277148,   email evertse at

Dr. Efthymios Sofos
office 238,  tel. 071-5277146,   email e.sofos at

Assistant (responsible for grading the homework):
Marc Paul Noordman,   office 207,  email marc.paul at

Time/Place: Mondays   13:45-15:30  Snellius 401
September 5-December 12, 2016, with the exception of October 3, November 7
EC points: 6
Examination: The examination consists of four homework assignments and a written exam in which you are asked questions about the contents of the course and about the homework exercises. Your grade for the exam, and the average of your grades of the four homework assignments, both contribute 50% to your final grade for Analytic number theory.

Possible resits will be given in the form of an oral exam.

Written exam: We have scheduled a written exam on Thursday January 26, 14-17 h. (2-5 pm), room 407/409.
Possible resits will be given in the form of individual oral exams.

In the written exam or oral resit we may ask questions about the homework exercises and variations thereof.
From the lecture notes we may ask questions about all theorems and about the proofs of the theorems and lemmas that are shorter than two pages and variations thereof, with the following exceptions:
- we will not ask questions about Chapter 0 (Prerequisites) but you have to know the results and be able to apply them;
- we will not ask questions about the proofs of Chapter 2 (Tools from complex analysis), but the theorems in that chapter are assumed to be known, in particular the (in)famous Theorem 2.6.
- we will not ask questions about §4.3 (Computation of G(q)); we may ask you to apply the results from §4.4 and §4.5, but you are not supposed to know the proofs in these sections;
- we will not ask questions about §5.4 (proof of the functional equation) but we may ask you to apply the functional equations for ζ(s) or L(s;χ). If so, we will recall the equation on the exam sheet;
- we will not ask questions about the proofs of the Tauberian theorems in Chapter 6, but we may ask you to formulate these theorems and you must be able to apply them.

Old exams: Exam January 22, 2015

Part of the course given last fall was given by Efthymios Sofos, and so its contents are partly different from those of the analytic number theory course given two years ago. You may also expect a question on the material treated by Efthymios.

Homework assignments (pdf): HOMEWORK GRADES

Homework assignment 1:   Due October 17
Homework assignment 2:   Due November 14
Homework assignment 3:   Due December 12
Homework assignment 4:   Due Sunday January 15

The deadlines for submitting your homework are strict.

Please do not forget to write or type your name and student number on your homework. To simplify grading it is preferable that you submit your homework in latex. Homework that is not well readable will not be graded. You may either deliver your homework at the course, give it to Marc Paul, or submit an electronic version to him by e-mail. In case you submit your homework by email, it should be by means of a single pdf-file, and typed in latex.

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 are not 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
Chapter 2: Tools from complex analysis
Chapter 3: Dirichlet series and arithmetic functions
Chapter 4: Characters and Gauss sums
Chapter 5: The Riemann zeta function and L-functions
Chapter 6: Tauberian theorems
Chapter 7: The Prime Number Theorem for arithmetic progressions
Chapter 8: Sums of nine positive cubes and the circle method
Chapter 9: Hua's lemma and a first expedition into the major arcs
Chapter 10: The singular integral
Chapter 11: The singular series

Remarks: This course will probably not be given in 2017/2018.
This course is recommended for a Master's thesis project in Number Theory.
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 (courses Analysis 1,2 and complex function theory in Leiden will do);
Algebra: abelian groups (small subset of course Algebra 1 in Leiden).
Literature: Recommended for further reading:
  • A. Granville, What is the best approach to counting primes, arXiv:1406.3754 [math.NT].
    Recent survey paper on estimates for π(x) (number of primes up to x) and related issues.

  • 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

  • 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

  • D.J. Newman, Analytic Number Theory, Springer Verlag, Gruduate 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
  • Useful
  • 2010 Mathematics Subject Classification (MSC2010)
    Official classification of mathematics subjects. The books in the Mathematical Institute library are classified according to this classification. The number theory books are 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 through the institute'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
    Mathematical preprints archive; number theory preprints are categorized under NT.
  • Contents (tentative, may be subject to change): The first part of this course will be on prime number theory. In our course we give rather recent, relatively simple proofs of both the Prime Number Theorem and the Prime Number Theorem for arithmetic progressions, which are due to Newman. In the second part we will discuss Waring's problem, which is about whether for each integer k≥2 there is a number g such that every positive integer can be expressed as a sum of at most g k-th powers. We intend to show that every sufficient large integer can be expressed as a sum of nine positive cubes. Below we give a short historical overview of the subject.

    Let π(x) denote the number of primes not exceeding x. In 1798, Legendre posed the following conjecture: π(x)∼x/log x as x→∞, that is, limx→∞π(x)(log x/x)=1. The first step in proving this conjecture was made by Chebyshev, who proved in 1851-52 that if limx→∞ π(x)(log x/x) exists, then the limit must be equal to 1. However, he was not able to prove the existence of the limit. The next important contribution was due to Riemann. In his famous memoir from 1859 (his only paper on number theory), he proved several results on the function ζ(s)=∑n≥1 n-s (now known as the Riemann zeta function) and stated several conjectures about this function. First observe that ζ(s) converges for all complex numbers s with real part larger than 1. Riemann showed that ζ(s) has a unique analytic continuation to ℂ\{1} (that is, there exists a unique analytic function on ℂ\{1} which coincides with ζ(s) on Re s >1), with a simple pole with residue 1 at s=1. Denoting this analytic continuation also by ζ(s), Riemann showed that ζ(s) satisfies a functional equation which relates ζ(s) to ζ(1-s). It is an easy consequence of this functional equation, that ζ(s) has simple zeros at all even negative integers -2,-4,-6,... and that all other zeros lie in the critical strip consisting of all complex numbers s with real part between 0 and 1. Riemann stated a still unproved famous conjecture, now known as the Riemann Hypothesis (RH): all zeros of ζ(s) in the critical strip lie on the axis of symmetry of the functional equation, that is on the line of all complex numbers with real part ½. In his memoir, Riemann mentioned several results, without proof or just with a sketch of a proof, relating the distribution of the zeros of ζ(s) to the distribution of the prime numbers. These results were proved completely in the 1890's by Hadamard and von Mangoldt. Finally, in 1896, Hadamard and de la Vallée Poussin independently proved Legendre's conjecture stated above, now known as the Prime Number Theorem. Their proofs were based on complex analysis, applied to the Riemann zeta function. Later, several other proofs of the Prime Number Theorem were given, all based on complex analysis, until in 1948 Erdős and Selberg independently published an "elementary proof" of the Prime Number Theorem, avoiding complex analysis.

    Dirichlet may be viewed as the founder of analytic number theory. In 1839-1842, he showed that each arithmetic progression a,a+q,a+2q,a+3q,... contains infinitely many prime numbers. In his proof he used properties of so-called L-functions L(s,χ)= ∑n≥1 χ(n)n-s where χ is a character modulo q. As it turned out, L-functions have many properties similar to those of the Riemann zeta function: for instance they have an analytic continuation to ℂ and they satisfy a functional equation. Further, there is a Generalized Riemann Hypothesis (GRH) which asserts that all zeros of (the analytic continuation of) L(s,χ) in the critical strip lie on the line of complex numbers with real part ½. De la Vallée Poussin proved the Prime Number Theorem for arithmetic progressions which is as follows: let π(x;a,q) denote the number of primes not exceeding x in the arithmetic progression a,a+q,a+2q,a+3q,... . Assume that 0<a<q and that a is coprime with q. 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 1770, Lagrange proved that every positive integer can be expressed as a sum of four squares. Subsequently, Waring claimed without proof that every positive integer is the sum of nine cubes, nineteen fourth powers, and so on. Denote by g(k) the smallest integer g such that every positive integer is expressable as a sum of g k-th powers, and by G(k) the smallest G such that every sufficiently large positive integer is expressable as a sum of G k-th powers, that is, all integers with the exception of at most finitely many can be expressed as a sum of G k-th powers. So G(k)≤g(k) for all k. In 1909, Hilbert proved that g(k) is finite for all k. Building further on work of Hardy and Ramanujan, in the early 1920's, Hardy and Littlewood developed a new method in analytic number theory, their famous circle method. Using this circle method, they gave the first estimates for G(k). Following the circle method, together with later refinements, it will be shown during our course that G(3)≤9.

      Back to top This website is maintained by Jan-Hendrik Evertse.