Primjena L-funkcija u teoriji brojeva

master's thesis

• Author: Marija Patljak
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-02-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu je dokazan Dirichletov teorem, rezultat koji dokazuje kako aritmetički niz oblika \(kn + h \quad (h, k)=1, \quad n=0,1,2, \dots, \) sadrži beskonačno mnogo prostih brojeva. U prvom poglavlju su dane osnovne definicije i svojstva funkcija koje koristimo u radu. Najvažniji rezultat izveden u ovom poglavlju je dokaz Harolda N. Shapira da red \(\Sigma \, p^{-1} \log p\) divergira, gdje se sumira po svim prostim brojevima \(p\). Također, dokazani su i osnovni teoremi koji pokazuju kako je skup prostih brojeva beskonačan. U drugom poglavlju se upoznajemo s karakterima konačne Abelove grupe. Od velike važnosti za dokaz Dirichletovog teorema su Dirichletovi karakteri \(\chi\) i funkcija \(L(1, \chi)\). Osobito je bitan dokaz kako je \(L(1, \chi) \neq 0\) za sve realne netrivijalne karaktere \(\chi\). Sam dokaz Dirichletovog teorema je izveden u trećem, ujedno i posljednjem, poglavlju. Nizom lema smo dokazali Dirichletov teorem kao posljedicu Shapirove formule (iz prvog poglavlja), proširene uvjetom na proste brojeve \(p\) kongruentne s \(h\) modulo \(k\).

Abstract (english)

This thesis proves Dirichlet’s theorem, a result that states that any arithmetic progression of the form \(kn + h \quad (h, k)=1, \quad n=0,1,2, \dots, \) contains infinitely many primes. In the first chapter we give basic definitions and properties of functions that are used in this thesis. The most important result derived in this section is the proof by Harold N. Shapiro that shows that the series \(\Sigma \, p^{-1} \log p\), extended over all primes, diverges. Also, the basic theorems that show that the set of prime numbers is infinite are proven. In the second chapter we become familiar with characters of finite Abelian groups. Of great importance for the proof of Dirichlet’s theorem are Dirichlet characters \(\chi\) and the functions \(L(1, \chi)\). It is particularly important to know that \(L(1, \chi) \neq 0\) for all real nonprincipal characters \(\chi\). The proof of Dirichlet’s theorem is carried out in the third, and the last, chapter. Through series of lemmas we prove Dirichlet’s theorem as a consequence of Shapiro’s formula (from the first chapter), extended over all primes \(p\) which are congruent to \(h\) mod \(k\).

• Keywords: Dirichletov teorem; Shapirova formula; konačne Abelove grupe
• Keywords (english): Dirichlet’s theorem; Shapiro’s formula; finite Abelian groups
• Language: croatian
• URN:NBN: urn:nbn:hr:217:907159
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-29 09:36:03