{"id":"https://openalex.org/W7128311193","doi":"https://doi.org/10.5281/zenodo.18526628","title":"Scale\u2013Localized Divisibility Rigidity and the Generalized Riemann Hypothesis","display_name":"Scale\u2013Localized Divisibility Rigidity and the Generalized Riemann Hypothesis","publication_year":2026,"publication_date":"2026-02-08","ids":{"openalex":"https://openalex.org/W7128311193","doi":"https://doi.org/10.5281/zenodo.18526628"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.18526628","doi":"https://doi.org/10.5281/zenodo.18526628","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.18526628","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},"volume":null,"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.18526628","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120913454","display_name":"Nigel Stafford Paltoo","orcid":"https://orcid.org/0009-0001-5580-7114","observed_orcids":["https://orcid.org/0009-0001-5580-7114"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paltoo, Nigel Stafford","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5121395012","display_name":"Eliyahu Fir-Sean Paltoo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paltoo, Eliyahu Fir-Sean","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5121353691","display_name":"Bindrawattie Singh-Paltoo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Singh-Paltoo, Bindrawattie","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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":0.18015696,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0,"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"}},{"id":"https://openalex.org/T12812","display_name":"Meromorphic and Entire Functions","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/generalized-riemann-hypothesis","display_name":"generalized Riemann hypothesis","score":0.9620000123977661},{"id":"https://openalex.org/keywords/dirichlet-l-functions","display_name":"Dirichlet L-functions","score":0.972000002861023},{"id":"https://openalex.org/keywords/admissibility-conditions","display_name":"admissibility conditions","score":0.7730000019073486},{"id":"https://openalex.org/keywords/euler-product","display_name":"Euler product","score":0.7099999785423279},{"id":"https://openalex.org/keywords/vertical-growth","display_name":"vertical growth","score":0.503000020980835},{"id":"https://openalex.org/keywords/critical-line","display_name":"critical line","score":0.5600000023841858},{"id":"https://openalex.org/keywords/nontrivial-zeros","display_name":"non-trivial zeros","score":0.8339999914169312},{"id":"https://openalex.org/keywords/dirichlet-characters","display_name":"Dirichlet characters","score":0.9039999842643738}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8913999795913696},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.6779000163078308},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.6080999970436096},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.5583000183105469},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5389000177383423},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5105999708175659},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.5030999779701233},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.4406000077724457},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.36059999465942383},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3571000099182129},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.35339999198913574},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.3504999876022339},{"id":"https://openalex.org/C2781217745","wikidata":"https://www.wikidata.org/wiki/Q1053779","display_name":"Euler system","level":3,"score":0.31220000982284546},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.30550000071525574},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.27149999141693115},{"id":"https://openalex.org/C187967104","wikidata":"https://www.wikidata.org/wiki/Q2264419","display_name":"Dirichlet L-function","level":5,"score":0.2694999873638153},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.26190000772476196}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.18526628","doi":"https://doi.org/10.5281/zenodo.18526628","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.18526628","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},{"id":"doi:10.5281/zenodo.18526629","doi":"https://doi.org/10.5281/zenodo.18526629","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.18526629","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.18526628","doi":"https://doi.org/10.5281/zenodo.18526628","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.18526628","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7138938570","https://openalex.org/W7203790866","https://openalex.org/W7153473795","https://openalex.org/W7117360319","https://openalex.org/W7134803232","https://openalex.org/W7161026450","https://openalex.org/W7128080834","https://openalex.org/W1987029514","https://openalex.org/W6949672088","https://openalex.org/W2950878008"],"abstract_inverted_index":{"We":[0,56],"develop":[1],"a":[2,12,20,117,147,159],"deterministic":[3],"framework":[4,98],"for":[5,70],"analyzing":[6],"weighted":[7],"prime":[8,87,109],"sums":[9,72,88],"based":[10],"on":[11,171],"logarithmic":[13,77],"decomposition":[14],"of":[15,86,93,106,120,125,131,164],"the":[16,97,107,121,129,155,172],"natural":[17],"numbers":[18],"and":[19,37,135,179,195],"forward":[21],"divisibility":[22],"principle.":[23],"Within":[24],"this":[25,60],"framework,":[26],"cancellation":[27],"at":[28],"fixed":[29],"scale":[30],"is":[31,80,111,177,180],"formalized":[32],"through":[33],"an":[34,66,113,132],"admissibility":[35,105,142],"condition":[36],"shown":[38,168],"to":[39,68,89,99,150,169],"be":[40,53],"rigid:":[41],"any":[42],"persistent":[43],"deviation":[44],"from":[45,154],"alternating":[46],"behavior":[47,85],"produces":[48],"scale-level":[49],"growth":[50,91,137],"that":[51,59,82,104],"cannot":[52],"neutralized":[54],"locally.":[55],"then":[57],"show":[58],"rigidity":[61,145,197],"propagates":[62],"across":[63],"scales,":[64],"yielding":[65],"obstruction":[67,149],"boundedness":[69],"prime-indexed":[71],"with":[73,143],"incompatible":[74],"weights.":[75],"A":[76],"truncation":[78],"principle":[79],"established":[81],"connects":[83],"scale-localized":[84],"vertical":[90,139],"properties":[92,124],"Euler":[94,133],"products.":[95],"Applying":[96],"Dirichlet":[100,126,165],"characters,":[101],"we":[102],"prove":[103],"associated":[108],"weights":[110],"not":[112],"additional":[114],"assumption,":[115],"but":[116],"necessary":[118],"consequence":[119],"classical":[122,192],"analytic":[123,151],"L-functions,":[127],"including":[128],"existence":[130],"product":[134],"polynomial":[136],"in":[138],"strips.":[140],"Combining":[141],"transported":[144],"yields":[146],"structural":[148],"vanishing":[152],"away":[153],"critical":[156,173],"line.":[157,174],"As":[158],"consequence,":[160],"all":[161],"nontrivial":[162],"zeros":[163],"L-functions":[166],"are":[167],"lie":[170],"The":[175],"argument":[176],"unconditional":[178],"carried":[181],"out":[182],"entirely":[183],"within":[184],"standard":[185],"set-theoretic":[186],"foundations,":[187],"using":[188],"only":[189],"elementary":[190],"arithmetic,":[191],"complex":[193],"analysis,":[194],"operator-theoretic":[196],"principles.":[198]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-02-09T00:00:00"}