ANALYTIC NUMBER THEORY

Mastermath course -  Fall 2020 - 8EC

[ Mastermath schedule  |  Mastermath webpage  |  Homework assignments  |  Course notes ]

Schedule: Classes: Thursdays   September 10-December 17, 14:00-16:45 (2-4:45 pm) Classes will be given on-line; details will follow.

Exam: Thursday  January 21, 2021, 14:00-17:00 (written, on-site or on-line)
Retake: Thursday  February 25, 2021, 14:00-17:00
(the retake will be replaced by an oral exam if the number of participants is small)

Teachers and assistants: Teachers:

Dr. Jan-Hendrik Evertse (Universiteit Leiden)
email evertse at math.leidenuniv.nl

Dr. Adelina Manzateanu (Universiteit Leiden)
email m.manzateanu at math.leidenuniv.nl

Assistants (responsible for the exercise classes and the grading of the homework): not yet known

Examination:

The examination will consist of some homework assignments and a written exam.
The average of the homework grades (rounded to one decimal) will contribute 30% to the final grade and the exam grade (rounded to one decimal) 70%. The final grade will be rounded to the nearest integer (≥0.5 up, <0.5 down).
To get a sufficient final grade (≥6), the grade for your written exam must be at least 5.0, irrespective of your homework grade.
Further details will follow.

Old exams: Exam 2014-2015 (covers only the material on prime number theory)
Exam 2016-2017 (covers everything)
Prerequisites: 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.
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 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. More precisely, we will give an asymptotic formula for the number of ways a given positive integer n can be expressed as a sum of nine positive cubes (i.e., with a main term and an error term of smaller order of magnitude). To achieve this we will employ the famous circle method, originating from Hardy, Ramanujan and Littlewood.

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. At least for the first part of the course, homework exercises will be selected from the exercises in the course notes.
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
Chapter 1: Introduction to prime number theory
Chapter 2: Dirichlet series and arithmetic functions
Chapter 3: Characters and Gauss sums
Chapter 4: The Riemann zeta function and L-functions
Chapter 5: Tauberian theorems
Chapter 6: The Prime Number Theorem for arithmetic progressions

This covers Jan-Hendrik's part of the course.

The notes covering Adelina's part will follow.

Remarks: This course will probably not be given in 2021/2022.
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 g non-negative k-th powers.The proofs are based on the circle method of Hardy and Littlewood.
    ISBN 0-521-60583-0

  • 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).

    Useful
    websites:
  • 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.
  • Historical overview: 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. This method has been successfully applied to give asymptotic formulas for the number of solutions of various types of Diophantine equations, for instance, to give an asymptotic formula for the number of times that a given positive integer n can be represented as a sum of G k-th powers, for G sufficiently large. Further, the circle method is the main tool in the solution of the ternary Goldbach problem (that every odd integer ≥7 is the sum of three primes), and it has been used to give a good asymptotic formula for the number of partitions of a given integer.

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