{"id":"https://openalex.org/W7204900407","doi":"https://doi.org/10.3934/math.2026927","title":"Square-free numbers of the form $ \\lfloor m^c\\rfloor $ with smooth indices","display_name":"Square-free numbers of the form $ \\lfloor m^c\\rfloor $ with smooth indices","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7204900407","doi":"https://doi.org/10.3934/math.2026927"},"language":"en","primary_location":{"id":"doi:10.3934/math.2026927","doi":"https://doi.org/10.3934/math.2026927","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.2026927","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/7/PDF/math-11-07-927.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"http://www.aimspress.com/aimspress-data/math/2026/7/PDF/math-11-07-927.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025408871","display_name":"Yanbo Song","orcid":"https://orcid.org/0009-0000-3643-4909","observed_orcids":["https://orcid.org/0009-0000-3643-4909"]},"institutions":[{"id":"https://openalex.org/I4210115513","display_name":"Xi\u2019an University","ror":"https://ror.org/01zzmf129","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210115513"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yanbo Song","raw_affiliation_strings":["Department of Mathematics, School of Artificial Intelligence and Big Data, Xi'an University, Xi'an 710065, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, School of Artificial Intelligence and Big Data, Xi'an University, Xi'an 710065, China","institution_ids":["https://openalex.org/I4210115513"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5025408871"],"corresponding_institution_ids":["https://openalex.org/I4210115513"],"apc_list":{"value":1200,"currency":"USD","value_usd":1200},"apc_paid":{"value":1200,"currency":"USD","value_usd":1200},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"11","issue":"7","first_page":"22999","last_page":"23015"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":9.999999747378752e-05,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/square-free-integers","display_name":"square-free integers","score":0.9900000095367432},{"id":"https://openalex.org/keywords/floor-function","display_name":"floor function","score":0.675000011920929},{"id":"https://openalex.org/keywords/exponential-sums","display_name":"exponential sums","score":0.9100000262260437},{"id":"https://openalex.org/keywords/integer-points-on-curves","display_name":"integer points on curves","score":0.5019999742507935},{"id":"https://openalex.org/keywords/diophantine-approximation","display_name":"Diophantine approximation","score":0.5320000052452087}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5910999774932861},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3215999901294708},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2924000024795532},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2809000015258789},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2696000039577484},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2418999969959259},{"id":"https://openalex.org/C116834253","wikidata":"https://www.wikidata.org/wiki/Q2039217","display_name":"Identification (biology)","level":2,"score":0.22920000553131104},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22609999775886536},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.22589999437332153},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.2222999930381775}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/math.2026927","doi":"https://doi.org/10.3934/math.2026927","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.2026927","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/7/PDF/math-11-07-927.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/math.2026927","doi":"https://doi.org/10.3934/math.2026927","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.2026927","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/7/PDF/math-11-07-927.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204900407.pdf","grobid_xml":"https://content.openalex.org/works/W7204900407.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1512958063","https://openalex.org/W1986514813","https://openalex.org/W2061950693","https://openalex.org/W2092182987","https://openalex.org/W2249264406","https://openalex.org/W2566336824","https://openalex.org/W3121630123","https://openalex.org/W3127767088","https://openalex.org/W3133264256","https://openalex.org/W4403995909","https://openalex.org/W4412644399","https://openalex.org/W4416115910","https://openalex.org/W4416972089","https://openalex.org/W6892431865","https://openalex.org/W7133237909","https://openalex.org/W7160774430","https://openalex.org/W7161063147"],"related_works":["https://openalex.org/W7204009524","https://openalex.org/W7212110517","https://openalex.org/W7098611450","https://openalex.org/W2010124539","https://openalex.org/W2388793102","https://openalex.org/W4286588328","https://openalex.org/W2272485272","https://openalex.org/W2207745004","https://openalex.org/W3080770573","https://openalex.org/W2319278212"],"abstract_inverted_index":{"The":[0,66],"aim":[1],"of":[2,11,28,31,41,47,84,86,107],"this":[3],"article":[4],"was":[5],"to":[6,38,70,95],"study":[7],"the":[8,12,39,45,51,82,105],"square-free":[9],"numbers":[10],"form":[13],"$":[14,17],"\\lfloor":[15],"m^c\\rfloor":[16],"with":[18,78],"smooth":[19],"indices,":[20],"which":[21,54],"can":[22,98],"be":[23,71,99],"seen":[24],"as":[25,92],"an":[26,36,56],"extension":[27,37],"a":[29,60,93],"result":[30],"Cao":[32,42,108],"and":[33,43,74,109,114],"Zhai.":[34,110],"As":[35],"work":[40,106],"Zhai,":[44],"proof":[46],"Theorem":[48,89,96,122],"1.1":[49,113],"employed":[50],"division":[52,67],"trick,":[53],"transformed":[55],"arithmetic":[57],"problem":[58,61],"into":[59],"concerning":[62],"exponential":[63],"sum":[64],"estimates.":[65],"trick":[68],"proved":[69,100],"more":[72],"direct":[73],"concise":[75],"in":[76],"comparison":[77],"alternative":[79],"approaches.":[80],"For":[81],"sake":[83],"completeness":[85],"our":[87,118],"results,":[88],"1.2,":[90,115],"serving":[91],"complement":[94],"1.1,":[97],"by":[101],"invoking":[102],"results":[103],"from":[104],"Combining":[111],"Theorems":[112],"we":[116],"obtained":[117],"full":[119],"conclusion,":[120],"namely,":[121],"1.3.":[123]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-09-02T00:00:00"}