Fareyevo zaporedje in Riemannova hipoteza

delo diplomskega seminarja

Tjaša Vrhovnik (Author), Aleš Vavpetič (Mentor)

Abstract

V delu je predstavljeno Fareyevo zaporedje, njegova motivacija in lastnosti. Navedena je rekurzivna formula za izračun dolžine zaporedja n-tega reda in njeno asimptotično obnašanje. Izkaže se, da je Fareyevo zaporedje v bijekciji z množico Fordovih krogov. Razložen je geometrijski pomen lastnosti mediante in Fareyevih sosedov ter opisana konstrukcija vseh Fordovih sosedov danega Fordovega kroga. Definirane so Fordove krogle. S pomočjo delovanja grupe Möbiusovih transformacij na množico Fordovih krogov je dokazana lastnost mediante. Predstavljena je Riemannova hipoteza in dve njeni ekvivalentni formulaciji. Dokazan je izrek, ki Fareyevo zaporedje preko Mertensove funkcije poveže z Riemannovo hipotezo.

Keywords

matematika;Fareyevo zaporedje;Fordov krog;zeta funkcija;Riemannova hipoteza;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[T. Vrhovnik]
UDC:	511
COBISS:	18724441
Views:	1111
Downloads:	203
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Farey Sequence and The Riemann Hypothesis
Secondary abstract:	The work presents the Farey sequence, its motivation and properties. A recursive formula for calculating the length of the sequence of order n and its asymptotic behaviour are stated. The Farey sequence is in bijection with the set of Ford circles, therefore a geometric meaning of the mediant and the neighbours property is explained as well as the construction of all Ford neighbours to a given Ford circle. Moreover, Ford spheres are defined. Using the fact that the group of Möbius transformations acts on the set of Ford circles, the mediant property for Ford circles is proved. Additionally, the Riemann hypothesis and two of its equivalent statements are presented. A theorem, connecting the Farey sequence, the Mertens function and the Riemann hypothesis, is proved.
Secondary keywords:	mathematics;Farey sequence;Ford circle;zeta function;Riemann hypothesis;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	33 str.
ID:	11227545