Riemannova funkcija zeta in praštevila

magistrsko delo

Petra Čačkov (Author), Marko Slapar (Mentor)

Abstract

V matematiki obstaja še veliko nedokazanih izrekov in trditev. Ena izmed njih je Riemannova hipoteza o ničlah funkcije zeta. Čeprav je funkcija dobila ime po matematiku Bernhardu Riemannu, jo je poznal že Euler približno 120 let pred njim. Ta je pokazal, da je funkcija zeta tesno povezana s praštevili. Euler je funkcijo zeta obravnaval nad realnimi števili, medtem ko jo je Riemann razširil tudi nad kompleksna števila. To je omogočilo uporabo drugih analitičnih orodij za raziskovanje praštevil. Funkcijo se s pomočjo funkcije gama in Möbiusove funkcije da analitično razširiti na celotno kompleksno ravnino in najbolj zanimiv del funkcije se nahaja v kritičnem pasu kompleksne ravnine. Riemannova hipoteza pravi, da imajo vse ničle funkcije, ki se nahajajo v tem pasu, realni del enak eni polovici. Hipoteze do danes ni dokazal še nihče, problem pa je uvrščen med probleme tisočletne nagrade Clayjevega matematičnega inštituta in je del Hilbertovega seznama 23 nerešenih problemaov (Hilbertov osmi problem). Riemannova hipoteza igra veliko vlogo pri preštevanju in porazdelitvi praštevil, saj ob predpostavki, da ta drži, dobimo “najboljšo možno” mejo odstopanja praštevilskega izreka.

Keywords

neskončne vrste;neskončni produkti;praštevila;praštevilski izrek;Riemannova hipoteza;Riemannova funkcija zeta;funkcija gama;Möbiusova funkcija;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[P. Čačkov]
UDC:	517.523(043.2)
COBISS:	12180041
Views:	629
Downloads:	128
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Riemann zeta function and prime numbers
Secondary abstract:	In mathematics still exist a lot of unproven theorems and one of them is Riemann hypothesis about zeroes of zeta function. Although the function was named after Bernhard Riemann, it was studied by Euler about 120 years before him. He showed that there exists a connection between zeta function and prime numbers. But Euler examined the function over real numbers while Riemann expended it to the whole complex plane which enabled the use of other analytical properties for the research of prime numbers. The zeta function can be analytically expanded over the whole complex plane with the help of gamma and Möbius function. The most interesting part of zeta function lies in critical strip of the complex plane. Riemann argued that all zeroes of the zeta function lying in the critical strip have the real part equal to one half. The hypothesis has never been proven and it is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems and is also one of the Clay Mathematics Institute's Millennium Prize Problems. The Riemann hypothesis plays an important part in the distribution as well as at counting the prime numbers. It implies "the best possible" bound for the error of the prime number theorem.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje, Predmetno poučevanje: Fizika in matematika
Pages:	65 str.
ID:	10978092