{"id":"https://openalex.org/W4210815915","doi":"https://doi.org/10.1080/00029890.2022.2026165","title":"The Powers of Certain Number-Theoretic Matrices","display_name":"The Powers of Certain Number-Theoretic Matrices","publication_year":2022,"publication_date":"2022-02-04","ids":{"openalex":"https://openalex.org/W4210815915","doi":"https://doi.org/10.1080/00029890.2022.2026165"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2022.2026165","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2022.2026165","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027680285","display_name":"Karl Dilcher","orcid":"https://orcid.org/0000-0002-9550-234X"},"institutions":[{"id":"https://openalex.org/I113979032","display_name":"Twitter (United States)","ror":"https://ror.org/04wt43v05","country_code":"US","type":"company","lineage":["https://openalex.org/I113979032"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Karl Dilcher","raw_affiliation_strings":["https://orcid.org/0000-0002-9550-234X"],"raw_orcid":"https://orcid.org/0000-0002-9550-234X","affiliations":[{"raw_affiliation_string":"https://orcid.org/0000-0002-9550-234X","institution_ids":["https://openalex.org/I113979032"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5060232837","display_name":"Kurt Girstmair","orcid":"https://orcid.org/0000-0003-3105-5111"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kurt Girstmair","raw_affiliation_strings":["View further author information"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"View further author information","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.02116484,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"129","issue":"4","first_page":"320","last_page":"335"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9937000274658203,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14500","display_name":"Advanced Mathematical Theories","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/eulers-totient-function","display_name":"Euler's totient function","score":0.9559735059738159},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.7127118110656738},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6947593092918396},{"id":"https://openalex.org/keywords/orthogonality","display_name":"Orthogonality","score":0.6101018786430359},{"id":"https://openalex.org/keywords/ramanujans-sum","display_name":"Ramanujan's sum","score":0.5937601923942566},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5885242819786072},{"id":"https://openalex.org/keywords/integer-matrix","display_name":"Integer matrix","score":0.5668052434921265},{"id":"https://openalex.org/keywords/greatest-common-divisor","display_name":"Greatest common divisor","score":0.5608216524124146},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.5438161492347717},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4751591384410858},{"id":"https://openalex.org/keywords/divisor","display_name":"Divisor (algebraic geometry)","score":0.4667031764984131},{"id":"https://openalex.org/keywords/number-theory","display_name":"Number theory","score":0.45744913816452026},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.4475197196006775},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4353610873222351},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.37791502475738525},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.1836576759815216},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10321193933486938},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08650597929954529},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07618266344070435}],"concepts":[{"id":"https://openalex.org/C198944804","wikidata":"https://www.wikidata.org/wiki/Q190026","display_name":"Euler's totient function","level":3,"score":0.9559735059738159},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7127118110656738},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6947593092918396},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.6101018786430359},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.5937601923942566},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5885242819786072},{"id":"https://openalex.org/C62555958","wikidata":"https://www.wikidata.org/wiki/Q6042601","display_name":"Integer matrix","level":5,"score":0.5668052434921265},{"id":"https://openalex.org/C125257309","wikidata":"https://www.wikidata.org/wiki/Q131752","display_name":"Greatest common divisor","level":2,"score":0.5608216524124146},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.5438161492347717},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4751591384410858},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.4667031764984131},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.45744913816452026},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.4475197196006775},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4353610873222351},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37791502475738525},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.1836576759815216},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10321193933486938},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08650597929954529},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07618266344070435},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C139018669","wikidata":"https://www.wikidata.org/wiki/Q6961560","display_name":"Nonnegative matrix","level":4,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2022.2026165","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2022.2026165","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321487","display_name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W883045577","https://openalex.org/W1994563393","https://openalex.org/W2016722805","https://openalex.org/W2068302185","https://openalex.org/W2147354174","https://openalex.org/W4252263778","https://openalex.org/W4292806204","https://openalex.org/W6754108485"],"related_works":["https://openalex.org/W1563683337","https://openalex.org/W62156077","https://openalex.org/W4383173604","https://openalex.org/W2034298439","https://openalex.org/W2463024015","https://openalex.org/W4289782853","https://openalex.org/W2947649114","https://openalex.org/W955074858","https://openalex.org/W4210815915","https://openalex.org/W2036976094"],"abstract_inverted_index":{"For":[0],"each":[1,30],"integer":[2,9,22],"n\u22653":[3],"we":[4],"define":[5],"a":[6,60],"specific":[7],"symmetric":[8],"matrix":[10],"of":[11,29,31,65],"size":[12],"\u03c6(n)/2.":[13],"We":[14],"then":[15],"explicitly":[16],"describe":[17],"all":[18],"positive":[19],"and":[20,38,78,84],"negative":[21],"powers,":[23,28],"as":[24,26],"well":[25],"fractional":[27],"these":[32],"matrices;":[33],"their":[34,73],"eigenvalues":[35],"also":[36],"appear":[37],"play":[39],"an":[40,48],"important":[41],"role.":[42],"All":[43],"this":[44,66],"is":[45],"based":[46],"on":[47],"orthogonality":[49],"relation":[50],"for":[51],"odd":[52],"Dirichlet":[53],"characters":[54],"which,":[55],"in":[56],"turn,":[57],"leads":[58],"to":[59],"diagonalization.":[61],"In":[62],"the":[63,76,85],"course":[64],"article,":[67],"various":[68],"other":[69],"number-theoretic":[70],"objects":[71],"make":[72],"appearance,":[74],"including":[75],"Euler":[77],"Jordan":[79],"totient":[80],"functions,":[81],"M\u00f6bius":[82],"inversion,":[83],"divisor":[86],"function.":[87]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}