p-adski brojevi i p-adska interpolacija Riemannove zeta funkcije

master's thesis

• Title (english): p-ad numbers and p-ad interpolation of the Riemann zeta function
• Author: Vanesa Babok
• Mentor: Marcela Hanzer (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj se diplomski rad bavi \(p\)-adskim brojevima i pruža svojevrstan uvod u \(p\)-adsku analizu. Cilj mu je definirati polje p-adskih brojeva \(\mathbb{Q}_p\) te postaviti temelje za usustavljivanje \(p\)-adskog analogona skupa kompleksnih brojeva \(\mathbb{C}\). Rad je podijeljen u tri poglavlja. Prvom poglavlje započinjemo prisjećanjem na pojmove polja, metrike, norme i metričkih prostora. Zatim uvodimo novu vrstu udaljenosti kojom ćemo se baviti kroz ostatak rada - \(p\)-adsku udaljenost. Također razlikujemo norme s obzirom na to jesu li arhimedske ili nearhimedske. Definiramo ekvivalentne norme i metrike i dokazujemo razna njihova svojstva te spominjemo teorem Ostrowskog. Potom se prisjećamo konstrukcije kompleksnih brojeva te na sličan način konstruiramo polje \(p\)-adskih brojeva \(\mathbb{Q}_p\). U ostatku poglavlja proučavamo njegovu aritmetiku i dokazujemo važnu Henselovu lemu. U drugom poglavlju proučavamo Riemannovu \(\zeta\) funkciju i njenu \(p\) -adsku interpolaciju. Započinjemo definiranjem Bernoullijevih brojeva \(B_k\) koji se javljaju u formuli za \(\zeta(2k)\) koju potom dokazujemo. Nadalje, promatramo topologiju na \(\mathbb{Q}_p\) te uvodimo kompaktno-otvorene skupove i lokalno konstantne funkcije. Njih zatim koristimo u definiciji \(p\)-adskih distribucija te navodimo nekoliko primjera istih. Dotičemo se i Bernoullijevih polinoma \(B_k(x) \) i Bernoullijevih distribucija, te uvodimo mjere i integraciju u \(\mathbb{Q}_p\). Koristeći uvedene distribucije i izraze za \(\zeta(2k)\), definiramo \(p\) -adske interpolacije \(\zeta\)-funkcije. U trećem poglavlju govorimo o gama funkciji. Dajemo njenu definiciju i dokaz njenih osnovnih svojstava. Pritom se prisjećamo nekih pojmova iz područja kompleksne analize poput analitičkih funkcija, polova i reziduuma. Poglavlje završavamo upotrebom gama funkcije pri dokazivanju formule za razvoj hiperbolnog sinusa u beskonačni produkt koja se bila spominjala u drugom poglavlju.

Abstract (english)

This thesis focuses on \(p\)-adic numbers and provides a sort of an introduction to the \(p\)-adic analysis. Its main goal is to define the field of \(p\)-adic numbers \(\mathbb{Q}_p\) and lay the foundation for introduction of the \(p\)-adic analogue of the set of complex numbers \(\mathbb{C}\). The paper is divided into three chapters. We begin the first chapter by recalling the concepts of fields, metrics, norms, and metric spaces. Then we introduce a new type of distance that we will deal with throughout the rest of the paper, namely \(p\)-adic distance. We also distinguish norms depending on whether they are Archimedean or non-Archimedean. We define equivalent norms and metrics, prove their various properties and mention the Ostrowski theorem. Then we recall the construction of complex numbers and in a similar way we construct the field of \(p\)-adic numbers \(\mathbb{Q}_p\). In the rest of the chapter we study its arithmetic and prove the important Hensel lemma. In the second chapter we study the Riemann \(\zeta\) function and its \(p\)-adic interpolation. We start by defining the Bernoulli numbers \(B_k\) that appear in the formula for\(\zeta(2k)\), which we then prove. Furthermore, we observe the topology on \(\mathbb{Q}_p\) and introduce compactopen sets and locally constant functions. We then use them in the definition of \(p\)-adic distributions and give a few examples. We also touch on Bernoulli polynomials \(B_k(x) \) and Bernoulli distributions, and introduce measures and integration in \(\mathbb{Q}_p\). Using the introduced distributions and expressions for \(\zeta(2k)\), we define \(p\)-adic interpolations of the \(\zeta\)-function. In the third chapter we talk about the gamma function. We give its definition and proof of its basic properties. In doing so, we recall some concepts from the field of complex analysis such as analytical functions, poles and residues. We conclude the chapter by using the gamma function in proving the infinite product identity for hyperbolic sine that was mentioned in the second chapter.

• Keywords: \(p\)-adska analiza; teorem Ostrowskog; Henselova lema; Riemannova \(\zeta\) funkcija; Bernoullijevi polinomi; Bernoullijeve distribucije; gama funkcija; razvoj hiperbolnog sinusa
• Keywords (english): \(p\)- adic analysis; Ostrowski theorem; Hensel lemma; Riemann \(\zeta\) function; Bernoulli polynomials; Bernoulli distributions; gamma function; hyperbolic sine
• Language: croatian
• URN:NBN: urn:nbn:hr:217:488675
• Study programme: Title: Mathematical Statistics; 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: 2022-03-25 12:12:51