NWI-WM139
Analytic Number Theory
Course infoSchedule
Course moduleNWI-WM139
Credits (ECTS)6
CategoryMA (Master)
Language of instructionEnglish
Offered byRadboud University; Faculty of Science; Wiskunde, Natuur- en Sterrenkunde;
Lecturer(s)
Coordinator
prof. dr. V. Zudilin
Other course modules lecturer
Lecturer
prof. dr. V. Zudilin
Other course modules lecturer
Contactperson for the course
prof. dr. V. Zudilin
Other course modules lecturer
Academic year2017
Period
KW1-KW2  (04/09/2017 to 04/02/2018)
Starting block
KW1
Course mode
full-time
Remarks-
Registration using OSIRISYes
Course open to students from other facultiesYes
Pre-registrationNo
Waiting listNo
Placement procedure-
Aims
  • The student has a sound theoretical knowledge of basic properties of Riemann's zeta function and Dirichlet's L-functions.
  • The student is familiar with complex analysis techniques for constructing analytic continuation of L-functions.
  • The student has a sound theoretical knowledge of results about approximation of real numbers by rationals.
  • The student has a sound theoretical knowledge of the Hermite-Padé approximation method for proving the transcendence of values of the exponential function.
Content
The course covers the three important topics in analytic number theory: (1) the prime number theorem (also known as the asymptotic distribution of primes); (2) Dirichlet's theorem about primes in arithmetic progressions; and (3) diophantine approximations and transcendence of e and π. En route we will learn analytic properties of Riemann's zeta function and, more generally, of Dirichlet's L-functions and gain information about approximation of real and complex numbers by rational/algebraic numbers. In spite of a classical character of this knowledge, the techniques and methods remain useful in contemporary research; some links with recent developments in analytic number theory will be provided.
Test information
The assessment for the course Analytic Number Theory is based on:

(1) One homework assignment on Topic 1 (the prime number theorem); problems are provided in Chapter 1 of the lecture notes.
Due: Tuesday 10 October 2017.

(2) One homework assignment on Topic 2 (Dirichlet's theorem on primes in arithmetic progressions); problems are provided in Chapter 2 of the lecture notes.
Due: Tuesday 14 November 2017.

(3) One homework assignment on Topic 3 (algebraic and transcendental numbers); problems are provided in Chapter 3 of the lecture notes.
Due: Tuesday 19 December 2017.

(4) Oral examination on all theoretical material covered. Each student will be requested to present a particular section from one of the chapters in full, and answer general questions discussed in the course.

If H1, H2, H3 and E are the marks (each of 10 maximal possible) then the final mark is computed as

max( (E+H1+H2+H3)/4, E/2+max(H1+H2,H2+H3,H3+H1)/4 )

and rounded to the nearest integer or half-integer.

Recommended materials
Book
Apostol, Tom M.: Introduction to analytic number theory, Undergraduate Texts in Mathematics (Springer-Verlag, New York-Heidelberg, 1976).
Book
Bateman, Paul T., and Diamond, Harold G.: Analytic number theory. An introductory course, Monographs in Number Theory 1 (World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2004).
Book
Newman, Donald J.: Analytic number theory, Graduate Texts in Mathematics 177 (Springer-Verlag, New York, 1998).
Book
Baker, Alan: Transcendental number theory, 2nd edition, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1990).

Instructional modes
Course
Attendance MandatoryYes

Tests
Tentamen
Test weight1
OpportunitiesBlock KW2, Block KW4