 Subjects(version: 850)
Course, academic year 2019/2020
Login via CAS Analytic and Combinatorial Number Theory - NDMI045
Title in English: Analytická a kombinatorická teorie čísel Department of Applied Mathematics (32-KAM) Faculty of Mathematics and Physics from 2018 summer 3 summer s.:2/0 Ex [hours/week] unlimited unlimited taught Czech, English full-time http://kam.mff.cuni.cz/~klazar/AKTC19.html
Guarantor: doc. RNDr. Martin Klazar, Dr. Informatika Mgr. - Diskrétní modely a algoritmyM Mgr. MSTRM Mgr. MSTR > Povinně volitelné Informatics > Discrete Mathematics
 Annotation - ---CzechEnglish
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 - ---CzechEnglish
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 - ---CzechEnglish
Last update: Mgr. Jan Kynčl, Ph.D. (31.05.2019)

Oral exam.

 Literature - ---CzechEnglish
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 - ---CzechEnglish
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 - ---CzechEnglish
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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