Fareyevo zaporedje in Riemannova hipoteza : delo diplomskega seminarja

Vrhovnik, Tjaša (avtor), Vavpetič, Aleš (mentor)

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.

• Jezik: Slovenski jezik
• Ključne besede: Fareyevo zaporedje, Fordov krog, zeta funkcija, Riemannova hipoteza
• Vrsta gradiva: Delo diplomskega seminarja/zaključno seminarsko delo/naloga
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2019
• PID: 20.500.12556/RUL-110587
• UDK: 511
• COBISS.SI-ID: 18724441
• Datum objave v RUL: 18.09.2019

Sekundarni jezik

• Jezik: Angleški jezik
• Naslov: The Farey Sequence and The Riemann Hypothesis

Izvleček

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.

• Ključne besede: Farey sequence, Ford circle, zeta function, Riemann hypothesis