Izreki Čebiševa in Mertensa o distribuciji praštevil

diplomsko delo

Mateja Adam (Author), Daniel Eremita (Mentor)

Abstract

V diplomskem delu so obravnavane aritmetične funkcije ter izreki Čebiševa in Mertensa o distribuciji praštevil. Prvo poglavje je namenjeno vpeljavi osnovnih pojmov in rezultatov, ki se uporabljajo skozi diplomsko delo. V drugem poglavju so predstavljene aritmetične funkcije, multiplikativne funkcije in Dirichletov produkt. Opisani sta dve metodi za ocenjevanje srednjih vrednosti aritmetičnih funkcij, integracija in parcialna sumacija. Izpeljana je tudi Möbiusova formula inverzije. Glavna tema diplomskega dela je obravnavana v tretjem poglavju. Prvi del tega poglavja je namenjen funkciji pi ter funkcijama Čebiševa theta in psi. Dokazano je, da imajo funkcije pi(x) ln x, theta(x) in psi(x) enak red velikosti kot funkcija x. V drugem delu tega poglavja so predstavljeni trije Mertensovi rezultati v zvezi s porazdelitvijo praštevil, med njimi je izpeljana tudi Mertensova formula.

Keywords

matematika;praštevila;distribucija;izreki;aritmetika;funkcije;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2011
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Adam]
UDC:	51(043.2)
COBISS:	18616328
Views:	2102
Downloads:	127
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	CHEBYSHEV'S AND MERTENS'S THEOREMS ON THE DISTRIBUTION OF PRIME NUMBERS
Secondary abstract:	The thesis considers arithmetic functions and classical theorems of Chebyshev and Mertens on the distribution of prime numbers. The first chapter introduces basic terms and results, which are used in the thesis. In the second chapter we consider arithmetic functions, multiplicative functions and the Dirichlet convolution. Two methods for estimating mean values of arithmetic functions, integration and partial summation are described. The Möbius inversion formula is also derived. The main topic of the thesis is considered in the third chapter. Its first part is focused on the function pi and the Chebyshev functions theta and psi. We derive the Chebyshev's theorem stating that the functions pi(x) ln x, theta(x) and psi(x) all have order of magnitude x. The second part of this chapter presents three Mertens's results regarding the distribution of prime numbers, among which the Mertens's formula is also derived.
Secondary keywords:	Arithmetic functions;prime numbers;distribution of prime numbers;Chebyshev's theorems;Mertens's theorems;Chebyshev functions;Mertens's formula;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	45 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	19494