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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kosonen, Kati", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-04-21T11:41:32Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-04-21T11:41:32Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/68629", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa osoitetaan, ett\u00e4 kaksi satunnaisesti valittua kokonaislukua ovat kesken\u00e4\u00e4n suhteellisia alkulukuja 61% todenn\u00e4k\u00f6isyydell\u00e4. Tulosta l\u00e4hestyt\u00e4\u00e4n lukuteorian n\u00e4k\u00f6kulmasta erilaisten funktioiden ja niiden ominaisuuksien avulla. Eulerin \\phi-funktio on merkitt\u00e4v\u00e4ss\u00e4 roolissa, sill\u00e4 tutkielman p\u00e4\u00e4tulos on Eulerin funktion keskim\u00e4\u00e4r\u00e4isen kasvunopeuden n\u00e4ytt\u00e4minen. T\u00e4m\u00e4n tuloksen sovelluksena pystyt\u00e4\u00e4n klassisen todenn\u00e4k\u00f6isyyden avulla osoittamaan alkulukuparien todenn\u00e4k\u00f6isyys. Tulos keskim\u00e4\u00e4r\u00e4iselle kasvunopeudelle on merkitt\u00e4v\u00e4 sen monipuolisten sovellusmahdollisuuksien takia. Tutkielmassa perehdyt\u00e4\u00e4n lukuteorian kahteen keskeiseen multiplikatiiviseen funktioon, Eulerin \\phi-funktioon ja M\u00f6biuksen \\mu-funktioon. K\u00e4yd\u00e4\u00e4n molempien funktioiden huomionarvoiset tulokset l\u00e4pi ja osoitetaan, miten funktiot ovat yhteydess\u00e4 toisiinsa. M\u00f6biuksen funktio on tutkielman t\u00e4rkeimpi\u00e4 ty\u00f6kaluja, koska sen yhteydet muihin tutkielmassa esitelt\u00e4viin funktioihin ovat p\u00e4\u00e4tuloksen kannalta olennaisia. Analyyttiseen lukuteoriaan syvennyt\u00e4\u00e4n tutkielman edetess\u00e4, kun k\u00e4sitell\u00e4\u00e4n funktiota \\zeta reaalisten arvojen tapauksessa. Eulerin \\zeta-funktio m\u00e4\u00e4ritell\u00e4\u00e4n sarjana, mutta se voidaan esitt\u00e4\u00e4 my\u00f6s p\u00e4\u00e4ttym\u00e4tt\u00f6m\u00e4n\u00e4 tulona. P\u00e4\u00e4ttym\u00e4tt\u00f6m\u00e4t tulot ovat tutkielman k\u00e4ytetyimpi\u00e4 ty\u00f6kaluja, joten perehdyt\u00e4\u00e4n niiden teoriaan tarkemmin. Funktioon \\zeta liittyy my\u00f6s tunnettu lukuteorian tulos, Baselin ongelma, jolle annetaan kaksi erilaista todistusta. Tutkielmassa tarkastellaan my\u00f6s toista Eulerin funktion nopeuden sovellusta. Toinen sovellus liittyy Fareyn jonoiksi kutsuttujen murtolukujonojen teoriaan, johon perehdyt\u00e4\u00e4n vuonna 1747 esitetyn kysymyksen saattelemana. Keskim\u00e4\u00e4r\u00e4isen kasvunopeuden tuloksen avulla pystyt\u00e4\u00e4n osoittamaan Fareyn jonojen asymptoottinen pituus. \nTutkielman lopuksi k\u00e4sitell\u00e4\u00e4n suppeasti kompleksianalyysin tuloksia sarjoille, jotta saadaan pohja esitell\u00e4 kompleksinen \\zeta-funktio ja sen nollakohdat. Kompleksisen \\zeta-funktion nollakohtien tarkasteluun liittyy vahvasti tunnetuin lukuteorian avoin ongelma, Riemannin hypoteesi. K\u00e4yd\u00e4\u00e4n l\u00e4pi millaisia l\u00e4hestymistapoja matemaatikoilla on ollut vuosien varrella hypoteesin todistamiseksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2020-04-21T11:41:32Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-04-21T11:41:32Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "61", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Baselin ongelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Riemannin hypoteesi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Fareyn jonot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "alkulukuparit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "multiplikatiivisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Suhteellisten alkulukuparien todenn\u00e4k\u00f6isyys", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202004212839", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "todenn\u00e4k\u00f6isyys", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukuteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "alkuluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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