{"id":"https://openalex.org/W2965714545","doi":null,"title":"In Half-Plane $\\text{Re}(s)\\le1$, Riemann's $\\zeta(s)$ is Convergent and the Dirichlet Series of $\\zeta(s)$ is Divergent, Violating the Law of Non-Contradiction","display_name":"In Half-Plane $\\text{Re}(s)\\le1$, Riemann's $\\zeta(s)$ is Convergent and the Dirichlet Series of $\\zeta(s)$ is Divergent, Violating the Law of Non-Contradiction","publication_year":2018,"publication_date":"2018-02-21","ids":{"openalex":"https://openalex.org/W2965714545","mag":"2965714545"},"language":"en","primary_location":{"id":"mag:2965714545","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1802.08062v6","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/abs/1802.08062v6","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080934429","display_name":"Ayal Sharon","orcid":"https://orcid.org/0000-0002-5690-6181","observed_orcids":["https://orcid.org/0000-0002-5690-6181"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ayal Sharon","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-5690-6181","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5080934429"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.9984999895095825,"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":0.9984999895095825,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.0003000000142492354,"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":0.00019999999494757503,"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/riemann-zeta-function","display_name":"Riemann zeta function","score":0.9459999799728394},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.9110000133514404},{"id":"https://openalex.org/keywords/dirichlet-series","display_name":"Dirichlet series","score":0.8379999995231628},{"id":"https://openalex.org/keywords/principle-of-non-contradiction","display_name":"principle of contradiction","score":0.9290000200271606},{"id":"https://openalex.org/keywords/law-of-excluded-middle","display_name":"law of excluded middle","score":0.9049999713897705},{"id":"https://openalex.org/keywords/three-valued-logic","display_name":"ternary logic","score":0.8809999823570251},{"id":"https://openalex.org/keywords/paradox","display_name":"paradox","score":0.38100001215934753},{"id":"https://openalex.org/keywords/branch-cuts","display_name":"branch cuts","score":0.4569999873638153},{"id":"https://openalex.org/keywords/cauchy-s-integral-theorem","display_name":"Cauchy's integral theorem","score":0.9110000133514404},{"id":"https://openalex.org/keywords/truth-value-gaps","display_name":"truth-value gaps","score":0.5049999952316284},{"id":"https://openalex.org/keywords/bernhard-riemann","display_name":"Bernhard Riemann","score":0.6129999756813049},{"id":"https://openalex.org/keywords/yang-mills-theory","display_name":"Yang-Mills theory","score":0.7260000109672546},{"id":"https://openalex.org/keywords/birch-swinnerton-dyer-conjecture","display_name":"Birch and Swinnerton-Dyer conjecture","score":0.7310000061988831}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.664391279220581},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.6207219958305359},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.4779089391231537},{"id":"https://openalex.org/C40758505","wikidata":"https://www.wikidata.org/wiki/Q386292","display_name":"Prime number theorem","level":3,"score":0.42060402035713196},{"id":"https://openalex.org/C82006148","wikidata":"https://www.wikidata.org/wiki/Q913764","display_name":"Cauchy's integral formula","level":4,"score":0.41482454538345337},{"id":"https://openalex.org/C204575570","wikidata":"https://www.wikidata.org/wiki/Q207476","display_name":"Holomorphic function","level":2,"score":0.4124700725078583},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3754951059818268},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3573235273361206},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3155757188796997},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.11519742012023926},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.09752002358436584},{"id":"https://openalex.org/C153635880","wikidata":"https://www.wikidata.org/wiki/Q1412436","display_name":"Cauchy problem","level":3,"score":0.07699641585350037}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:2965714545","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1802.08062v6","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null}],"best_oa_location":{"id":"mag:2965714545","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1802.08062v6","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},"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/W2792825188","https://openalex.org/W2903530165","https://openalex.org/W4394668781","https://openalex.org/W2788816558","https://openalex.org/W2964474121","https://openalex.org/W3156625358","https://openalex.org/W3084315925","https://openalex.org/W2971966010","https://openalex.org/W2788112650","https://openalex.org/W3156665291"],"abstract_inverted_index":{"Propositional":[0],"logics":[1],"and":[2,58,71,115,131,156,181,201,208,226,234,242,247,267],"Three-Valued":[3],"Logics":[4],"(3VL)":[5],"are":[6,152,161,165,183,190],"useful":[7],"tools":[8],"for\nanalyzing":[9],"the":[10,28,39,64,85,102,128,163,250],"Riemann":[11],"Hypothesis":[12],"(RH).":[13],"Riemann's":[14,80,88],"expression":[15,31],"of":[16,32,41,69,87,96,135],"$\\zeta(s)$\nclaims":[17],"to":[18,77,93,100,106,215,230],"be":[19,48,216,231],"convergent":[20,49],"throughout":[21],"half-plane":[22,56],"$\\text{Re}(s)\\le1$":[23],"(except":[24],"at\n$s=1$).":[25],"Given":[26],"that":[27,45,79,179,188,197],"Dirichlet":[29],"series":[30],"$\\zeta(s)$":[33,52,81,89,172],"is":[34,53,82,90,113,118,194,255,259],"proven":[35],"to\nbe":[36],"divergent":[37],"there,":[38],"Law":[40],"Non-Contradiction":[42],"(LNC)":[43],"holds":[44,178,187,228],"$\\zeta(s)$\ncannot":[46],"also":[47,263],"there.":[50],"So":[51,62,192],"exclusively":[54],"divergent\nthroughout":[55],"$\\text{Re}(s)\\le1$,":[57],"has":[59,173],"no":[60,174],"zeros.":[61],"$\\zeta(1)\\ne0$,\nresolving":[63],"BSD":[65],"conjecture":[66],"in":[67],"favor":[68],"finiteness,":[70],"2D":[72],"Yang-Mills":[73],"theory\nis":[74],"invalid,":[75,91],"due":[76,92],"assuming":[78],"true.":[168],"Also,":[84],"derivation":[86],"Riemann's\nimproper":[94],"use":[95],"Cauchy's":[97,136],"integral":[98,137],"theorem":[99],"equate":[101],"branch":[103,111,129],"cut":[104,112,130],"of\n$f(x)=\\log(-x)$":[105],"a":[107,195],"Hankel":[108,132],"contour.":[109],"The":[110],"non-holomorphic,":[114],"the\nHankel":[116],"contour":[117,133],"open":[119],"(or":[120],"closed":[121],"at":[122],"infinity,":[123],"thereby":[124],"encircling\nnon-holomorphic":[125],"points).":[126],"Both":[127],"contradict\nprerequisites":[134],"theorem.":[138],"In":[139,169,204,236,245],"set":[140],"theory,":[141],"$\\zeta(s)$'s":[142],"zeros":[143,151,160],"form":[144],"an":[145],"empty":[146],"set,":[147],"so":[148],"RH":[149,180,193,214,229],"(all":[150],"on\nthe":[153],"critical":[154],"line)":[155],"$\\overline{RH}$":[157,182],"(not":[158],"all":[159],"on":[162],"critical\nline)":[164],"both":[166,184,189,232],"vacuously":[167],"classical":[170],"logic,":[171,238],"zeros,\nso":[175],"material":[176],"implication":[177],"true,":[185,256],"and\nconjunction":[186],"false.":[191,220,235],"paradox":[196],"violates":[198],"LNC":[199,225],"and\nLEM,":[200],"causes":[202],"ECQ.":[203,271],"contrast,":[205],"Frege's,":[206],"{\\L}ukasiewicz's,":[207,262],"Kleene's":[209,246],"3VLs":[210],"reject":[211],"LEM,":[212],"and\nhold":[213],"neither":[217],"true":[218,233,260],"nor":[219],"Priest's":[221,248],"3VL":[222],"further":[223],"rejects":[224],"ECQ,\nand":[227],"Frege's":[237],"truth-value":[239],"gaps\nimply":[240],"gluts,":[241],"vice":[243],"versa.":[244],"3VLs,":[249],"proposition":[251],"if\nRH,":[252],"then":[253],"$X$":[258,265],"if":[257,264],"(in":[261],"is\nboth),":[266],"does":[268],"not":[269],"cause":[270]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2019-08-13T00:00:00"}