401-3100-68L  Introduction to Analytic Number Theory

SemesterAutumn Semester 2018
LecturersI. N. Petrow
Periodicitynon-recurring course
Language of instructionEnglish



Courses

NumberTitleHoursLecturers
401-3100-68 GIntroduction to Analytic Number Theory4 hrs
Wed08:15-10:00HG E 22 »
Thu15:15-17:00HG D 1.1 »
I. N. Petrow

Catalogue data

AbstractThis course is an introduction to classical multiplicative analytic number theory. The main object of study is the distribution of the prime numbers in the integers. We will study arithmetic functions and learn the basic tools for manipulating and calculating their averages. We will make use of generating series and tools from complex analysis.
ObjectiveThe main goal for the course is to prove the prime number theorem in arithmetic progressions: If gcd(a,q)=1, then the number of primes p = a mod q with p<x is approximately (1/phi(q))*(x/log x), as x tends to infinity, where phi(q) is the Euler totient function.
ContentDeveloping the necessary techniques and theory to prove the prime number theorem in arithmetic progressions will lead us to the study of prime numbers by Chebyshev's method, to study techniques for summing arithmetic functions by Dirichlet series, multiplicative functions, L-series, characters of a finite abelian group, theory of integral functions, and a detailed study of the Riemann zeta function and Dirichlet's L-functions.
Lecture notesLecture notes will be provided for the course.
LiteratureMultiplicative Number Theory by Harold Davenport
Multiplicative Number Theory I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan
Analytic Number Theory by Henryk Iwaniec and Emmanuel Kowalski
Prerequisites / NoticeComplex analysis
Group theory
Linear algebra
Familiarity with the Fourier transform and Fourier series preferable but not required.

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits8 credits
ExaminersI. N. Petrow
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationwritten 180 minutes
Additional information on mode of examinationThe examination will only be offered twice, in the Winter 2019 and Summer 2019 examination sessions.
Written aidsSheet of formulas provided by the lecturer. No student-produced written aids.
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

Learning materials

 
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Offered in

ProgrammeSectionType
Mathematics BachelorSelection: Algebra, Number Thy, Topology, Discrete Mathematics, LogicWInformation
Mathematics MasterSelection: Algebra, Number Thy, Topology, Discrete Mathematics, LogicWInformation