20.500.12556/RUL-110587 Fareyevo zaporedje in Riemannova hipoteza delo diplomskega seminarja The Farey Sequence and The Riemann Hypothesis 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. 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. Fareyevo zaporedje Fordov krog zeta funkcija Riemannova hipoteza Farey sequence Ford circle zeta function Riemann hypothesis true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2019-09-18 07:45:22 2019-09-18 07:45:24 2024-05-30 10:23:15 0000-00-00 00:00:00 2019 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511 99756 18724441 1071.pdf 1071.pdf 1 B158AA0DAD62365D83444984C6D334CD 7b2a6280d1858e27a0377139895e8f449a27262e575483f5637c8003fd09419f 26ea5c1f-a1b7-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=122260 Fakulteta za matematiko in fiziko 0 0 0