{"id":"https://openalex.org/W7221564321","doi":"https://doi.org/10.48550/arxiv.2610.12816","title":"Reciprocal LCM powers and Robin's criterion","display_name":"Reciprocal LCM powers and Robin's criterion","publication_year":2026,"publication_date":"2026-10-08","ids":{"openalex":"https://openalex.org/W7221564321","doi":"https://doi.org/10.48550/arxiv.2610.12816"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.12816","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12816","pdf_url":"https://arxiv.org/pdf/2610.12816","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.12816","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016942284","display_name":"Grant Molnar","orcid":"https://orcid.org/0000-0001-9152-2477","observed_orcids":["https://orcid.org/0000-0001-9152-2477"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Molnar, Grant","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5016942284"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":1.0,"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":1.0,"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/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.0,"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"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.7770000100135803},{"id":"https://openalex.org/keywords/divisor-function","display_name":"divisor function","score":0.41100001335144043},{"id":"https://openalex.org/keywords/mobius-function","display_name":"M\u00f6bius function","score":0.7929999828338623},{"id":"https://openalex.org/keywords/robin-inequality","display_name":"Robin inequality","score":0.7990000247955322},{"id":"https://openalex.org/keywords/average-order","display_name":"average order","score":0.7580000162124634},{"id":"https://openalex.org/keywords/interval-arithmetic","display_name":"interval arithmetic","score":0.6859999895095825}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8384000062942505},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.7890999913215637},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6759999990463257},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.6119999885559082},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.5033000111579895},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.45010000467300415},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4311000108718872},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3910999894142151},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3864000141620636},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.34599998593330383},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3285999894142151},{"id":"https://openalex.org/C191252586","wikidata":"https://www.wikidata.org/wiki/Q1671453","display_name":"Interval arithmetic","level":3,"score":0.31610000133514404},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3127000033855438},{"id":"https://openalex.org/C84135550","wikidata":"https://www.wikidata.org/wiki/Q3154022","display_name":"Linear inequality","level":3,"score":0.2883000075817108},{"id":"https://openalex.org/C102607178","wikidata":"https://www.wikidata.org/wiki/Q102761","display_name":"Least common multiple","level":2,"score":0.28519999980926514},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.25600001215934753},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.12816","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12816","pdf_url":"https://arxiv.org/pdf/2610.12816","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.12816","doi":"https://doi.org/10.48550/arxiv.2610.12816","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.12816","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.12816","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12816","pdf_url":"https://arxiv.org/pdf/2610.12816","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221564321.pdf","grobid_xml":"https://content.openalex.org/works/W7221564321.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2377757523","https://openalex.org/W4294827068","https://openalex.org/W4416435938","https://openalex.org/W7172503835","https://openalex.org/W7097944991","https://openalex.org/W2581491121","https://openalex.org/W2060160466","https://openalex.org/W2951411482","https://openalex.org/W4225304578","https://openalex.org/W7095759752"],"abstract_inverted_index":{"For":[0],"$\u03ba>":[1],"1$,":[2],"we":[3],"study":[4],"the":[5,43,58],"reciprocal":[6],"LCM-power":[7],"function":[8],"\\[":[9],"\u03c3^{[\u03ba]}_{-1}(n)":[10],":=":[11],"\\sum_{d":[12],"\\mid":[13],"n}\u03bc(d)\\,\u03c3_{-1}(n":[14],"/":[15],"d)^\u03ba.":[16],"\\]":[17],"We":[18],"determine":[19],"its":[20],"average":[21],"and":[22,34,65],"extremal":[23],"orders.":[24],"A":[25],"natural":[26],"Robin-scale":[27],"normalization":[28],"is":[29],"strictly":[30],"increasing":[31],"in":[32],"$\u03ba$":[33],"tends":[35],"to":[36],"$\u03c3_{-1}$,":[37],"yielding":[38],"an":[39],"RH-equivalent":[40],"criterion.":[41],"Unconditionally,":[42],"corresponding":[44],"Robin":[45],"inequality":[46,59],"holds":[47],"eventually":[48],"on":[49],"every":[50],"bounded":[51],"parameter":[52],"interval.":[53],"Directed":[54],"interval":[55],"arithmetic":[56],"proves":[57],"for":[60],"each":[61],"$n":[62],">":[63],"5040$":[64],"$1":[66],"<":[67],"\u03ba\\le":[68],"100{,}000$.":[69]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-13T00:00:00"}
