{"id":"https://openalex.org/W7214044645","doi":"https://doi.org/10.48550/arxiv.2609.22759","title":"Distributional Fractional Taylor Series and Interpretation of the Fractal and Number-Theoretic Explicit Formulas","display_name":"Distributional Fractional Taylor Series and Interpretation of the Fractal and Number-Theoretic Explicit Formulas","publication_year":2026,"publication_date":"2026-09-19","ids":{"openalex":"https://openalex.org/W7214044645","doi":"https://doi.org/10.48550/arxiv.2609.22759"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.22759","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.22759","pdf_url":"https://arxiv.org/pdf/2609.22759","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/2609.22759","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5042734167","display_name":"Michel L. Lapidus","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I103635307","display_name":"University of California, Riverside","ror":"https://ror.org/03nawhv43","country_code":"US","type":"education","lineage":["https://openalex.org/I103635307"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lapidus, Michel L.","raw_affiliation_strings":["|| University of California , Riverside , Department of Mathematics , 900 University Ave , Riverside , CA 92521-0135 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"|| University of California , Riverside , Department of Mathematics , 900 University Ave , Riverside , CA 92521-0135 , USA","institution_ids":["https://openalex.org/I103635307"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152492331","display_name":"Matthew Overduin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Overduin, Matthew","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"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":0.6977999806404114,"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":0.6977999806404114,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.10490000247955322,"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.04610000178217888,"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/fractal-strings","display_name":"fractal strings","score":0.6909999847412109},{"id":"https://openalex.org/keywords/dirac-delta-function","display_name":"Dirac delta function","score":0.7009999752044678},{"id":"https://openalex.org/keywords/fractional-derivatives","display_name":"fractional derivatives","score":0.5339999794960022},{"id":"https://openalex.org/keywords/mellin-transform","display_name":"Mellin transform","score":0.8479999899864197},{"id":"https://openalex.org/keywords/laurent-series","display_name":"Laurent series","score":0.6790000200271606},{"id":"https://openalex.org/keywords/error-term","display_name":"error term","score":0.4560000002384186}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8220999836921692},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.7706000208854675},{"id":"https://openalex.org/C184311908","wikidata":"https://www.wikidata.org/wiki/Q425432","display_name":"Laurent series","level":2,"score":0.7099000215530396},{"id":"https://openalex.org/C83764196","wikidata":"https://www.wikidata.org/wiki/Q17014577","display_name":"Fractal derivative","level":5,"score":0.6590999960899353},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45910000801086426},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.4345000088214874},{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.4198000133037567},{"id":"https://openalex.org/C199874217","wikidata":"https://www.wikidata.org/wiki/Q606934","display_name":"Mellin transform","level":3,"score":0.3984000086784363},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39410001039505005},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3801000118255615},{"id":"https://openalex.org/C153363305","wikidata":"https://www.wikidata.org/wiki/Q1306887","display_name":"Geometric series","level":3,"score":0.3659000098705292},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36500000953674316},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.3637000024318695},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.3474999964237213},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.34209999442100525},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3407999873161316},{"id":"https://openalex.org/C186080144","wikidata":"https://www.wikidata.org/wiki/Q358733","display_name":"Series expansion","level":2,"score":0.33889999985694885},{"id":"https://openalex.org/C60396315","wikidata":"https://www.wikidata.org/wiki/Q5477489","display_name":"Fractal dimension on networks","level":5,"score":0.31380000710487366},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.31200000643730164},{"id":"https://openalex.org/C96175551","wikidata":"https://www.wikidata.org/wiki/Q3100949","display_name":"Fractal landscape","level":5,"score":0.30570000410079956},{"id":"https://openalex.org/C30573924","wikidata":"https://www.wikidata.org/wiki/Q905326","display_name":"Digamma function","level":5,"score":0.289900004863739},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.27469998598098755},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.26930001378059387}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.22759","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.22759","pdf_url":"https://arxiv.org/pdf/2609.22759","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.2609.22759","doi":"https://doi.org/10.48550/arxiv.2609.22759","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.22759","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.22759","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.22759","pdf_url":"https://arxiv.org/pdf/2609.22759","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":[{"id":"https://openalex.org/G6062635606","display_name":"Fractal Research And Computing Team with Applications in Life Sciences","funder_award_id":"ANR-24-CE45-3362","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320322892","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214044645.pdf","grobid_xml":"https://content.openalex.org/works/W7214044645.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1565489864","https://openalex.org/W1969341747","https://openalex.org/W2003880880","https://openalex.org/W2056513508","https://openalex.org/W2058171746","https://openalex.org/W2077158529","https://openalex.org/W2093660071","https://openalex.org/W2093876395","https://openalex.org/W2138242926","https://openalex.org/W2162388724","https://openalex.org/W2485490691","https://openalex.org/W2485578081","https://openalex.org/W2622140534","https://openalex.org/W2795326229","https://openalex.org/W4238350589","https://openalex.org/W4280516550","https://openalex.org/W4297803238","https://openalex.org/W4390870024","https://openalex.org/W4404920839"],"related_works":["https://openalex.org/W6981915650","https://openalex.org/W4234573985","https://openalex.org/W2963763583","https://openalex.org/W2963478164","https://openalex.org/W1993576382","https://openalex.org/W1969341747","https://openalex.org/W2998878163","https://openalex.org/W3035725599","https://openalex.org/W2896719952","https://openalex.org/W4285045073"],"abstract_inverted_index":{"Motivated":[0],"by":[1,91,107,120,184,239],"the":[2,49,66,83,92,96,108,137,148,163,167,173,179,192,202,205,220,223,227,233,240,254],"work":[3],"of":[4,56,65,95,104,111,139,178,191,204,208,219,226,232,257,262],"M.":[5,9],"L.":[6],"Lapidus":[7],"and":[8,33,147],"van":[10],"Frankenhuisjen":[11],"on":[12,75],"fractal":[13,18,27,37,43,60,68,98,150,263,270],"explicit":[14,46,151,170],"formulas":[15,47,171],"for":[16,41],"generalized":[17,42,59,97],"strings,":[19],"we":[20,161],"develop":[21],"in":[22,63,136,166,212,217,222,249,260],"this":[23,159,213,250],"paper":[24,251],"a":[25,54,58,88,112],"distributional":[26,36,169,176],"Taylor's":[28],"formula":[29,152],"with":[30],"error":[31,128,143],"term":[32,129,144,165],"an":[34,127],"exact":[35],"Taylor":[38,264],"series":[39,265],"representation":[40],"strings.":[44],"The":[45,210,246],"that":[48,134],"aforementioned":[50],"authors":[51],"developed":[52],"provide":[53],"way":[55],"expressing":[57],"string":[61],"$\u03b7$":[62],"terms":[64,218,261],"underlying":[67,269],"complex":[69,93,271],"dimensions,":[70],"under":[71],"suitable":[72],"growth":[73],"assumptions":[74],"its":[76],"geometric":[77,234],"zeta":[78,235],"function,":[79],"$\u03b6_\u03b7$,":[80,105,237],"known":[81],"as":[82,172],"languidity":[84,123],"assumptions.":[85],"This":[86],"involves":[87],"sum":[89,214],"indexed":[90],"dimensions":[94],"string,":[99],"summing":[100],"over":[101],"all":[102],"residues":[103],"multiplied":[106,238],"Mellin":[109],"transform":[110],"tempered":[113],"or":[114],"Schwartz":[115],"test":[116,189],"function":[117,236,242],"$\u03d5$,":[118],"denoted":[119,183],"$\\widetilde\u03d5$.":[121],"Under":[122,131],"assumptions,":[124,133],"there":[125],"is":[126,145,153],"present.":[130],"stronger":[132],"is,":[135],"case":[138],"strong":[140],"languidity,":[141],"no":[142],"present":[146],"resulting":[149],"said":[154],"to":[155,187,201,253],"be":[156],"exact.":[157],"In":[158],"paper,":[160],"express":[162],"residue":[164],"fractional":[168,174],"$(-\u03c9)^{\\mathrm{th}}$-ordered":[175],"derivative":[177],"Dirac":[180],"$\u03b4$":[181],"distribution,":[182],"$Y_\u03c9$,":[185],"applied":[186],"modified":[188],"functions":[190],"form":[193],"$\u03d5(x)":[194],"\\ln^{k-1}(x)$,":[195],"where":[196],"$k$":[197],"runs":[198],"from":[199],"$1$":[200],"multiplicity":[203],"pole":[206],"$\u03c9$":[207,231],"$\u03b6_\u03b7$.":[209],"coefficients":[211,221],"are":[215],"written":[216],"principal":[224],"part":[225],"Laurent":[228],"expansion":[229],"at":[230,244],"gamma":[241],"evaluated":[243],"$\u03c9$.":[245],"results":[247],"obtained":[248],"contribute":[252],"broader":[255],"program":[256],"characterizing":[258],"fractals":[259],"expansions":[266],"involving":[267],"their":[268],"dimensions.":[272]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}