Prosti brojevi u aritmetičkim nizovima

master's thesis

• Title (english): Prime numbers in arithmetic sequences
• Author: Igor Škoro
• Mentor: Marko Tadić (mentor)
• Committee member: Marko Tadić (predsjednik povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-10-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu se proučavaju prosti brojevi u aritmetičkim nizovima. Glavni cilj rada je pokazati koji artimetički nizovi sadrže beskonačno mnogo prostih brojeva. U prvom poglavlju, navesti ćemo osnovne definicije i dokaze iz polja algebre, teorije brojeva i kompleksne analize, koji će nam trebati u ostatku rada. U drugom poglavlju cilj nam je dokazati Dirichletov teorem o prostim brojevima, koji govori da aritmetički niz ak+m, k ∈ \(\mathbb{N}\) sadrži beskonačno mnogo prostih brojeva ukoliko su a i m relativno prosti. U ovom poglavlju, prvo ćemo razmotriti nekoliko aritmetičkih nizova, za koje je jednostavno pokazati da sadrže beskonačno mnogo prostih brojeva. Zatim ćemo uvesti pojam Karaktera konačne Abelove grupe, Dirichletovih redova, L-funkcija i Dirichletove gustoće, koji će nam biti potrebni da onda dokažemo Dirichletov teorem. Na kraju, navesti ćemo primjenu Dirichletovog teorema u teoriji brojeva.

Abstract (english)

In this thesis we examine prime numbers in arithmetic progressions. The main purpose of the thesis is to show which arithmetic progressions contain infinitely many prime numbers. In the first chapter, we will state definitions and theorems from the fields of algebra, number theory and complex analysis, which we will use in the rest of the thesis. In the second chapter, our goal is to prove Dirichlet’s theorem on arithmetic progression, which states that the arithmetic progression ak + m, k ∈ \(\mathbb{N}\), contains infinitely many primes if a and m are relatively prime. In this chapter, we will first consider a few arithmetic progressions, for which it is easy to show that they contain infinitely many primes. Next, we will introduce the ideas of group characters, Dirichlet series, L-functions and Dirichlet density, which we will then use to prove Dirichlet’s theorem. Finally, we will show an application of Dirichlet’s theorem in number theory.

• Keywords: algebra; teorija brojeva; kompleksna analiza; Dirichletov teorem o prostim brojevima; Karakter konačne Abelove grupe; Dirichletovi redovi; L-funkcija; Dirichletova gustoća
• Keywords (english): algebra; number theory; complex analysis; Dirichlet's theorem on arithmetic progression; ideas of group characters; Dirichlet series; L-function; Dirichlet density
• Language: croatian
• URN:NBN: urn:nbn:hr:217:566452
• Study programme: Title: Pure 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: 2022-11-09 12:12:19