SubjectsSubjects(version: 850)
Course, academic year 2019/2020
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Analytic and Combinatorial Number Theory - NDMI045
Title in English: Analytická a kombinatorická teorie čísel
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information:
Guarantor: doc. RNDr. Martin Klazar, Dr.
Class: Informatika Mgr. - Diskrétní modely a algoritmy
M Mgr. MSTR > Povinně volitelné
Classification: Informatics > Discrete Mathematics
Annotation -
Last update: T_KAM (27.04.2005)
The course will cover some classic as well as some recent results of analytic and combinatorial number theory.
Aim of the course -
Last update: T_KAM (20.04.2008)

Students learn several fundamental results of analytic and combinatorial number theory and get familiar with the corresponding techniques.

Course completion requirements -
Last update: Mgr. Jan Kynčl, Ph.D. (31.05.2019)

Oral exam.

Literature -
Last update: T_KAM (20.04.2008)

G. Tenenbaum: Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press 1995.

Further references will be given in the lecture.

Requirements to the exam -
Last update: doc. RNDr. Martin Klazar, Dr. (11.06.2019)

Exam is oral, with written preparation. For concrete requirements see the above homepage of the course. Examples of exam questions:

1. Give general overview (without going into details) of the proof of Dirichlet's theorem on primes in arithmetic progression.

2. Prove Schur's asymptotics for the restricted partition function p_A(n).

3. Prove an upper bound on the partition function, of the form p(n) <= e^{cn^{1/2}}(either by induction or by Euler's generating function).

Syllabus -
Last update: T_KAM (27.04.2005)

The course will consist of a selection of the following topics. Prime number theorem. Dirichlet's theorem on primes in arithmetic progressions. Irrationality of zeta(3). Introduction to modular forms. Shnirelman's theorem on primes and Selberg's sieve. Vinogradov's three primes theorem. Freiman's theorem in additive number theory. T. Tao's proof of Szemeredi's theorem on arithmetic progressions, ...

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