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[{"key": "dc.contributor.advisor", "value": "Kurittu, Lassi", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Ylinen, Henri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-09-06T09:41:25Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-09-06T09:41:25Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1573506", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/51220", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutustuttaa lukija Bernoullin polynomeihin,\r\n\u0393-funktioon ja lukuteoreettisiin Mertensin lauseisiin. N\u00e4iden lis\u00e4ksi tutkitaan er\u00e4st\u00e4\r\nlukuteoreettista tuloa, ja esitell\u00e4\u00e4n t\u00e4h\u00e4n tuloon liittyvi\u00e4 tiett\u00e4v\u00e4sti uusia tuloksia.\r\n\r\nBernoullin polynomien avulla todistetaan erityisesti Euler-Maclaurinin lause, joka\r\nkertoo erilaisten summien ja integraalien v\u00e4lisest\u00e4 yhteydest\u00e4. \u0393-funktion avulla taas\r\ntodistetaan Stirlingin kaava, joka antaa hyv\u00e4n approksimaation kertoman n! kasvu-\r\nnopeudesta. Mertensin lauseista ensimm\u00e4inen kertoo, miten nopeasti lukua n pie-\r\nnempien alkulukujen k\u00e4\u00e4nteislukujen 1/p summa hajaantuu, kun kasvatetaan lukua n.\r\nToinen Mertensin lause kertoo, kuinka nopeasti kohti nollaa menee samojen alkulu-\r\nkujen yli otettava tulo, jossa kerrotaan kesken\u00e4\u00e4n lukuja 1 \u2212 1/p.\r\n\r\nN\u00e4iden ty\u00f6kalujen avulla tutkitaan lopuksi er\u00e4st\u00e4 tuloa. Kiinte\u00e4lle kokonaisluvulle n\r\nt\u00e4ss\u00e4 tulossa kerrotaan kesken\u00e4\u00e4n ne luvut 1 \u2264 k \u2264 n, joilla ei ole yhteisi\u00e4 tekij\u00f6it\u00e4\r\nluvun n kanssa. T\u00e4lle tulolle saadaan tiett\u00e4v\u00e4sti uusia yl\u00e4- ja alarajoja, joista osa on\r\ntarkkoja. N\u00e4iden rajojen etuna on se, ettei niiden laskemiseen tarvita tietoa luvuista\r\nk, vaan riitt\u00e4\u00e4 tiet\u00e4\u00e4 luvun n alkutekij\u00e4hajotelma.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Henri Ylinen (hejoylin) on 2016-09-06 09:41:24.916742. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-09-06T09:41:25Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201609063990.pdf: 490881 bytes, checksum: 07d4c91adc0f99b9890c0f5bcc8527d3 (MD5)\r\nlicense.html: 4784 bytes, checksum: 9f60af3ce56efe62211b23c5ee4e4034 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-09-06T09:41:25Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201609063990.pdf: 490881 bytes, checksum: 07d4c91adc0f99b9890c0f5bcc8527d3 (MD5)\r\nlicense.html: 4784 bytes, checksum: 9f60af3ce56efe62211b23c5ee4e4034 (MD5)\r\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (71 sivua)", "language": null, "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": "Bernoullin polynomit", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Bernoulli", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Gamma", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Gamma-funktio", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Mertens", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Mertensin lause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "phi-torial", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Reaalianalyyttist\u00e4 lukuteoriaa", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", 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"value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "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": "dc.date.updated", "value": "2016-09-06T09:41:26Z", "language": "", "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": 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