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Primes and L-functions - NMAG473

Title:	Prvočísla a L-funkce
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2025
Semester:	summer
E-Credits:	3
Hours per week, examination:	summer s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	English, Czech
Teaching methods:	full-time
Additional information:	https://sites.google.com/view/shman/modular-forms-and-l-functions-ii-summer-2324

Guarantor:	doc. Mgr. Vítězslav Kala, Ph.D.
Teacher(s):	Stylianos Sachpazis, Dr.
Class:	M Mgr. MSTR M Mgr. MSTR > Povinně volitelné
Classification:	Mathematics > Algebra

Opinion survey results SS schedule Noticeboard

Annotation -

L-functions are central objects in modern number theory, which played an important role in the proof of Fermat's Last Theorem. They are certain complex functions encoding information of number-theoretic interest, e.g., about the distribution of prime numbers. The course will cover their basic properties, especially concerning the existence and distribution of primes. Specific choice of topics will depend on the interests of participants.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (03.06.2025)

Literature -

• T. M. Apostol, Introduction to Analytic Number Theory, UTM, Springer-Verlag, 1976

• H. Davenport, Multiplicative Number Theory, 3rd edition, GTM 74, Springer-Verlag, 2000

• D. Koukoulopoulos, The Distribution of Prime Numbers, GSM 203, 2019

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (05.12.2025)

Requirements to the exam - Czech

Zkouška bude ústní s 30-60 minutami na přípravu jedné nebo dvou otázek, odpovídajících probrané látce na přednáškách.

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (14.02.2018)

Syllabus -

Arithmetic functions

Averages of arithmetic functions

Chebyshev's bounds and Mertens' theorems

Dirichlet series

Perron formulas

Riemann zeta function

Prime Number Theorem

Dirichlet's theorem on the infinitude of primes in arithmetic progressions

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (05.12.2025)