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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Still, Ida", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-05-21T11:09:51Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-05-21T11:09:51Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/75853", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on esitt\u00e4\u00e4 todistus neuvostoliittolaisen matemaatikon Lev Schnirelmannin 1930-luvun alussa osoittamalle tulokselle, jonka mukaan on olemassa sellainen luku S, ett\u00e4 jokainen ykk\u00f6st\u00e4 suurempi luonnollinen luku voidaan esitt\u00e4\u00e4 alkulukujen summana, jossa summattavia on korkeintaan S kappaletta. Schnirelmannin lause on merkitt\u00e4v\u00e4 additiivisen lukuteorian tulos, sill\u00e4 sen avulla p\u00e4\u00e4stiin l\u00e4hemm\u00e4ksi yht\u00e4 lukuteorian ratkaisematonta ongelmaa, Goldbachin konjektuuria. Ty\u00f6n alussa osoitetaan alkulukujen tiheyteen liittyv\u00e4t Mertensin lauseet, joita tarvitaan Schnirelmannin lauseen todistamiseen. Mertensin lauseista ensimm\u00e4inen kertoo, miten lukua x pienempien alkulukujen k\u00e4\u00e4nteislukujen 1/p summa k\u00e4ytt\u00e4ytyy, kun x kasvaa rajatta. Mertensin toinen lause puolestaan kertoo saman lukujen (1-1/p) tulolle. Schnirelmannin lauseeseen liittyv\u00e4t olennaisesti seulamenetelm\u00e4t, joilla voidaan arvioida kokoa positiivisista kokonaisluvuista koostuvalle, tietyt ehdot t\u00e4ytt\u00e4v\u00e4lle seulotulle joukolle. Yleinen ongelma, johon seuloja hy\u00f6dynnet\u00e4\u00e4n, on muotoa: Jos A on \u00e4\u00e4rellinen joukko positiivisia kokonaislukuja ja P \u00e4\u00e4rellinen joukko alkulukuja, niin kuinka paljon on sellaisia joukon A alkioita, jotka eiv\u00e4t ole jaollisia mill\u00e4\u00e4n alkuluvulla joukosta P? Seulamenetelmien yleisen idean lis\u00e4ksi ty\u00f6ss\u00e4 tutustutaan tarkemmin norjalaisen matemaatikon Viggo Brunin kehitt\u00e4m\u00e4\u00e4n seulaan, jota hy\u00f6dynt\u00e4m\u00e4ll\u00e4 saadaan osoitettua muutama Schnirelmannin lauseen todistamiseen tarvittava aputulos. N\u00e4m\u00e4 tulokset liittyv\u00e4t siihen, kuinka monella eri tavalla luonnollinen luku voidaan esitt\u00e4\u00e4 kahden alkuluvun summana. Ty\u00f6n t\u00e4rkeimpi\u00e4 k\u00e4sitteit\u00e4 on Schnirelmannin tiheys, jolla voidaan mitata luonnollisten lukujen osajoukon kokoa. Ty\u00f6ss\u00e4 k\u00e4yd\u00e4\u00e4n l\u00e4pi Schnirelmannin tiheyden ominaisuuksia ja osoitetaan tulos, jonka avulla voidaan arvioida Schnirelmannin tiheytt\u00e4 joukkojen summalle. Lis\u00e4ksi osoitetaan, ett\u00e4 jokainen ep\u00e4negatiivisista kokonaisluvuista koostuva joukko A, joka sis\u00e4lt\u00e4\u00e4 nollan ja jonka Schnirelmannin tiheys on positiivinen, on jonkin kertaluvun kanta ep\u00e4negatiivisten kokonaislukujen joukolle. T\u00e4ll\u00e4 tarkoitetaan sit\u00e4, ett\u00e4 on olemassa luonnollinen luku h siten, ett\u00e4 jokainen ep\u00e4negatiivinen kokonaisluku voidaan esitt\u00e4\u00e4 sellaisena joukon A alkioiden summana, jossa summattavia on h kappaletta. T\u00e4t\u00e4 tulosta hy\u00f6dynt\u00e4en esitet\u00e4\u00e4n lopuksi todistus Schnirelmannin lauseelle.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-05-21T11:09:51Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-05-21T11:09:51Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "53", "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": "additiivinen lukuteoria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "seulamenetelm\u00e4t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Schnirelmannin lause", "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-202105213113", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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": "alkuluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukuteoria", "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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