{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23444"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23444","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On the coefficients of cyclotomic polynomials","abstract":"Let $\\Phi\\sb{n}(z)$ denote the $n$th cyclotomic polynomial, given by$$\\Phi\\sb{n}(z) = {\\prod\\limits\\sbsp{a=1\\atop(a,n)=1}{n}}\\ (z - \\exp(2\\pi ia/n)) = {\\sum\\limits\\sbsp{m=0}{\\phi(n)}} a(m,n)z\\sp{m}.$$It is easily verified that for $n > 1$ $$\\Phi\\sb{n}(z)={\\prod\\limits\\sb{d\\vert n}}(1-z\\sp{d})\\sp{\\mu(n/d)},$$where $\\mu$ is the Moebius function. Hence the coefficients $a(m,n)$ of $\\Phi\\sb{n}(z)$ are integers, and for every fixed $m,\\ a(m,n)$ assumes only finitely many possible values.","abstract_html":"Let $\\Phi\\sb{n}(z)$ denote the $n$th cyclotomic polynomial, given by$<span class=\"etd-inline-math\">\\Phi\\sb{n}(z) = {\\prod\\limits\\sbsp{a=1\\atop(a,n)=1}{n}} (z - \\exp(2&pi; ia/n)) = {\\sum\\limits\\sbsp{m=0}{\\phi(n)}} a(m,n)z\\sp{m}.</span>$It is easily verified that for $n &gt; 1$ $<span class=\"etd-inline-math\">\\Phi\\sb{n}(z)={\\prod\\limits\\sb{d\\vert n}}(1-z\\sp{d})\\sp{&mu;(n/d)},</span>$where $\\mu$ is the Moebius function. Hence the coefficients $a(m,n)$ of $\\Phi\\sb{n}(z)$ are integers, and for every fixed $m,\\ a(m,n)$ assumes only finitely many possible values.","abstract_has_math":true,"creators":["Bachman, Gennady"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hildebrand, A.J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:14:24Z","date_published":"2011-05-07T14:14:24Z","updated_at":"2026-07-22T22:25:22Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1991 Bachman, Gennady"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9210733","(UMI)AAI9210733"],"render_values":[{"text":"AAI9210733","href":null,"code":true},{"text":"(UMI)AAI9210733","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23444","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hildebrand, A.J."]},{"key":"dc:creator","label":"Author","values":["Bachman, Gennady"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:14:24Z","10000-01-01","1991"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1991 Bachman, Gennady"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9210733","(UMI)AAI9210733","http://hdl.handle.net/2142/23444"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let $\\Phi\\sb{n}(z)$ denote the $n$th cyclotomic polynomial, given by$$\\Phi\\sb{n}(z) = {\\prod\\limits\\sbsp{a=1\\atop(a,n)=1}{n}}\\ (z - \\exp(2\\pi ia/n)) = {\\sum\\limits\\sbsp{m=0}{\\phi(n)}} a(m,n)z\\sp{m}.$$It is easily verified that for $n > 1$ $$\\Phi\\sb{n}(z)={\\prod\\limits\\sb{d\\vert n}}(1-z\\sp{d})\\sp{\\mu(n/d)},$$where $\\mu$ is the Moebius function. Hence the coefficients $a(m,n)$ of $\\Phi\\sb{n}(z)$ are integers, and for every fixed $m,\\ a(m,n)$ assumes only finitely many possible values.","We consider here the behavior of the function$$a(m) = {\\max\\limits\\sb{n}}\\ \\vert a(m,n)\\vert.$$Our principal result is an asymptotic formula for log $a(m)$ with logarithmic error term that improves over a recent estimate of Montgomery and Vaughan. We also give similar formulae for the logarithms of the one-sided extrema $a\\sp* (m)$ = max$\\sb{n}\\ a(m,n)$ and $a\\sb*(m)$ = min$\\sb{n}\\ a(m,n).$ In the course of the proof we obtain estimates for certain exponential sums which are of independent interest.","Made available in DSpace on 2011-05-07T14:14:24Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210733.pdf: 2153806 bytes, checksum: 23e409ee3380d74f38ba8d419e1947ce (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:04:31Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:30:50-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["On the coefficients of cyclotomic polynomials"]}]}],"canonical_facts":{"dc:contributor":["Hildebrand, A.J."],"dc:creator":["Bachman, Gennady"],"dc:date":["2011-05-07T14:14:24Z","10000-01-01","1991"],"dc:description":["Let $\\Phi\\sb{n}(z)$ denote the $n$th cyclotomic polynomial, given by$$\\Phi\\sb{n}(z) = {\\prod\\limits\\sbsp{a=1\\atop(a,n)=1}{n}}\\ (z - \\exp(2\\pi ia/n)) = {\\sum\\limits\\sbsp{m=0}{\\phi(n)}} a(m,n)z\\sp{m}.$$It is easily verified that for $n > 1$ $$\\Phi\\sb{n}(z)={\\prod\\limits\\sb{d\\vert n}}(1-z\\sp{d})\\sp{\\mu(n/d)},$$where $\\mu$ is the Moebius function. Hence the coefficients $a(m,n)$ of $\\Phi\\sb{n}(z)$ are integers, and for every fixed $m,\\ a(m,n)$ assumes only finitely many possible values.","We consider here the behavior of the function$$a(m) = {\\max\\limits\\sb{n}}\\ \\vert a(m,n)\\vert.$$Our principal result is an asymptotic formula for log $a(m)$ with logarithmic error term that improves over a recent estimate of Montgomery and Vaughan. We also give similar formulae for the logarithms of the one-sided extrema $a\\sp* (m)$ = max$\\sb{n}\\ a(m,n)$ and $a\\sb*(m)$ = min$\\sb{n}\\ a(m,n).$ In the course of the proof we obtain estimates for certain exponential sums which are of independent interest.","Made available in DSpace on 2011-05-07T14:14:24Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210733.pdf: 2153806 bytes, checksum: 23e409ee3380d74f38ba8d419e1947ce (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:04:31Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:30:50-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9210733","(UMI)AAI9210733","http://hdl.handle.net/2142/23444"],"dc:language":["eng"],"dc:rights":["Copyright 1991 Bachman, Gennady"],"dc:subject":["Mathematics"],"dc:title":["On the coefficients of cyclotomic polynomials"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:22Z"}