ANALYTIC NUMBER THEORY

Mastermath course -  Fall 2022 - 8EC

[ Mastermath schedule  |  Mastermath webpage  |  Course notes ]

Schedule: 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.
The retake will be replaced by an oral exam if the number of participants is small; such an oral exam will take about one hour.

Teachers and assistants: Teachers:
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 homework):
Sebastián Carrillo Santana (Universiteit Utrecht)  email: s.carrillosantana at uu.nl
Mike Daas (Universiteit Leiden)  email: m.a.daas at math.leidenuniv.nl

Questions:

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.
If you have a (private, non-mathematical) question for one of the teachers, you can best send him/her an email.
If you have a question for the assistants, e.g., about the grading of the homework, you can best send them an email.

Examination:

The examination will consist of twelve weekly homework assignments and a written exam.
The homework assignments will be posted on the ELO-page of Analytic Number Theory.
Only the 10 assignments with the highest grades will be counted. The average of these (rounded to one decimal) will contribute 25% to the final grade and the exam grade (rounded to one decimal) 75%. 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.

Homework assignments:
  • Homework assignments and their deadlines of delivery will be posted on the ELO page. Please note that these deadlines are strict.
    At least for the first part of the course, homework exercises will be selected from the exercises in the course notes.
  • You may submit your homework either in (la)tex, which we prefer, or handwritten, but in case you submit it in handwritten form it should be very well readable and have no erasures. The teaching assistants who do the grading have the right to ignore your written answers if they cannot read them. Do not forget to write (very well readable) or type your name, university and student number on your homework.
  • Homework should be uploaded on the ELO-page of Analytic Number Theory, in the form of a single pdf file (so in case you want to submit it in handwritten form, you shouldn't make pictures of the separate pages, but you should make one single pdf. There are many pdf-scan apps for smartphones).
Old exams: 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)
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, 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, limx→∞π(x)(log x/x)=1. 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):
Jan-Hendrik's part:
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.

At a request of a student I have uploaded the other chapters of part 1 of the course, but these may be subject to small changes.

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

Remarks: This course will probably not be given in 2023/2024.
This course is recommended for a Master's thesis project in Number Theory.
Literature: Recommended for further reading:
  • A. Cojocaru and M. R. Murty, An Introduction to Sieve Methods and Their Applications, Cambridge University Press; 1st edition (January 30, 2006)
    This is a very readable book on sieve methods.
    ISBN 0-521-61275-6

  • 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

  • P. Pollack, Not Always Buried Deep, American Mathematical Society; New edition (October 14, 2009)
    This book focuses on elementary proofs of results from analytic number theory (largely avoiding complex analysis).
    ISBN 0-821-84880-1

  • 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:
  • 2020 Mathematics Subject Classification (MSC2020)
    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-... . 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.

  • 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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