{"id":"https://openalex.org/W7223710126","doi":"https://doi.org/10.48550/arxiv.2610.32694","title":"A genus-two spectral curve of the Riemann $\u039e$-function: Real Structure and Certified Arithmetic Separation of Periods","display_name":"A genus-two spectral curve of the Riemann $\u039e$-function: Real Structure and Certified Arithmetic Separation of Periods","publication_year":2026,"publication_date":"2026-10-23","ids":{"openalex":"https://openalex.org/W7223710126","doi":"https://doi.org/10.48550/arxiv.2610.32694"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.32694","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32694","pdf_url":"https://arxiv.org/pdf/2610.32694","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/2610.32694","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014291916","display_name":"Michel R. P. Planat","orcid":"https://orcid.org/0000-0001-5739-546X","observed_orcids":["https://orcid.org/0000-0001-5739-546X"]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I2802759292","display_name":"Franche-Comt\u00e9 \u00c9lectronique M\u00e9canique Thermique et Optique - Sciences et Technologies","ror":"https://ror.org/004fmxv66","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I2802759292","https://openalex.org/I37553959","https://openalex.org/I4210095849","https://openalex.org/I4405256580","https://openalex.org/I53262699"]},{"id":"https://openalex.org/I4405256580","display_name":"Universit\u00e9 Marie et Louis Pasteur","ror":"https://ror.org/04asdee31","country_code":"FR","type":"education","lineage":["https://openalex.org/I4405256580"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Planat, Michel","raw_affiliation_strings":["Institut FEMTO-ST , CNRS UMR 6174 , Universit\u00e9 Marie et Louis Pasteur , Besan\u00e7on , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut FEMTO-ST , CNRS UMR 6174 , Universit\u00e9 Marie et Louis Pasteur , Besan\u00e7on , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I2802759292","https://openalex.org/I4405256580"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5014291916"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I2802759292","https://openalex.org/I4405256580"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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":0.9717000126838684,"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":0.9717000126838684,"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":0.01269999984651804,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.0032999999821186066,"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-xi-function","display_name":"Riemann xi function","score":0.8880000114440918},{"id":"https://openalex.org/keywords/spectral-curves","display_name":"spectral curves","score":0.6290000081062317},{"id":"https://openalex.org/keywords/genus-2-curves","display_name":"genus 2 curves","score":0.8740000128746033},{"id":"https://openalex.org/keywords/period-ratio","display_name":"period ratio","score":0.6169999837875366},{"id":"https://openalex.org/keywords/interval-arithmetic","display_name":"interval arithmetic","score":0.6669999957084656},{"id":"https://openalex.org/keywords/painleve-iii","display_name":"Painlev\u00e9 III","score":0.878000020980835},{"id":"https://openalex.org/keywords/kummer-surfaces","display_name":"Kummer surfaces","score":0.7020000219345093},{"id":"https://openalex.org/keywords/de-bruijn-newman-constant","display_name":"de Bruijn-Newman constant","score":0.9010000228881836},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.859000027179718},{"id":"https://openalex.org/keywords/isomonodromic-deformations","display_name":"isomonodromic deformations","score":0.8399999737739563},{"id":"https://openalex.org/keywords/heat-equation","display_name":"heat equation","score":0.4300000071525574},{"id":"https://openalex.org/keywords/farey-fractions","display_name":"Farey fractions","score":0.5149999856948853},{"id":"https://openalex.org/keywords/picard-rank","display_name":"Picard rank","score":0.6899999976158142},{"id":"https://openalex.org/keywords/transcendence","display_name":"transcendence","score":0.41100001335144043}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8877999782562256},{"id":"https://openalex.org/C206530611","wikidata":"https://www.wikidata.org/wiki/Q905896","display_name":"Diophantine equation","level":2,"score":0.5516999959945679},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.47099998593330383},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.4595000147819519},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.459199994802475},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.40450000762939453},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.3986000120639801},{"id":"https://openalex.org/C40621787","wikidata":"https://www.wikidata.org/wiki/Q16251580","display_name":"Diophantine set","level":3,"score":0.3837999999523163},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38179999589920044},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.3815999925136566},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.37070000171661377},{"id":"https://openalex.org/C112343008","wikidata":"https://www.wikidata.org/wiki/Q140744","display_name":"Arithmetic function","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.3626999855041504},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.3587000072002411},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3504999876022339},{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.3416000008583069},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.3273000121116638},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.3041999936103821},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.30219998955726624},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.29499998688697815},{"id":"https://openalex.org/C2776061190","wikidata":"https://www.wikidata.org/wiki/Q7451805","display_name":"Separation (statistics)","level":2,"score":0.2944999933242798},{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.2831000089645386},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.27649998664855957},{"id":"https://openalex.org/C191252586","wikidata":"https://www.wikidata.org/wiki/Q1671453","display_name":"Interval arithmetic","level":3,"score":0.26510000228881836}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.32694","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32694","pdf_url":"https://arxiv.org/pdf/2610.32694","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.2610.32694","doi":"https://doi.org/10.48550/arxiv.2610.32694","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.32694","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:2610.32694","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32694","pdf_url":"https://arxiv.org/pdf/2610.32694","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223710126.pdf","grobid_xml":"https://content.openalex.org/works/W7223710126.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W100965798","https://openalex.org/W102194274","https://openalex.org/W203922851","https://openalex.org/W1490126868","https://openalex.org/W1537838966","https://openalex.org/W1764048552","https://openalex.org/W1995434240","https://openalex.org/W1997454240","https://openalex.org/W2014664484","https://openalex.org/W2031865554","https://openalex.org/W2055369887","https://openalex.org/W2061097380","https://openalex.org/W2066411678","https://openalex.org/W2101991307","https://openalex.org/W2150722482","https://openalex.org/W2161221308","https://openalex.org/W2161468011","https://openalex.org/W2520295258","https://openalex.org/W2552835512","https://openalex.org/W2940545432","https://openalex.org/W2952101351","https://openalex.org/W2962680933","https://openalex.org/W3016082228","https://openalex.org/W3018101995","https://openalex.org/W3124379896","https://openalex.org/W7162654015","https://openalex.org/W7165674161","https://openalex.org/W7171636682","https://openalex.org/W7214984124","https://openalex.org/W7220973347","https://openalex.org/W7221785903"],"related_works":["https://openalex.org/W3121963155","https://openalex.org/W7221146789","https://openalex.org/W7135385164","https://openalex.org/W7167918243","https://openalex.org/W7223773933","https://openalex.org/W7128396328","https://openalex.org/W7117356479","https://openalex.org/W2964297629","https://openalex.org/W7221781177","https://openalex.org/W7212792009"],"abstract_inverted_index":{"We":[0],"associate":[1],"a":[2,9,24,94,100,120,131],"compact":[3],"genus-two":[4],"real":[5,25,42],"spectral":[6],"core":[7,137],"with":[8,105],"transverse":[10],"logarithmic-derivative":[11],"discriminant":[12],"of":[13,40,63],"the":[14,70,87,113],"Riemann":[15],"$\u039e$-function.":[16],"Directed":[17],"interval":[18],"arithmetic":[19,125],"certifies":[20],"its":[21],"branch":[22],"geometry,":[23],"$P_{\\mathrm{III}}(D_6)$":[26],"scalar-normal-form":[27],"realization,":[28],"and":[29,32,53,65,76,108,126,139],"associated":[30],"Kummer":[31],"Shioda--Inose":[33],"models.":[34],"Zero-action":[35],"resonance":[36],"reduces":[37],"to":[38],"rationality":[39],"two":[41],"period":[43],"ratios.":[44],"Their":[45],"certified":[46],"heat":[47,71,83],"motion":[48],"yields":[49],"generic":[50],"local":[51,95,124],"nonresonance":[52,138],"Diophantine":[54],"separation,":[55],"while":[56],"explicit":[57],"Farey":[58],"windows":[59],"impose":[60],"denominator":[61],"barriers":[62],"$1{,}036{,}003$":[64],"$140{,}769$":[66],"throughout":[67],"$|\u03ba|\\le3\\times10^{-10}$":[68],"in":[69],"time":[72],"$\u03ba$.":[73],"Fixed-point":[74],"irrationality":[75],"exact":[77],"Picard":[78],"rank":[79],"remain":[80],"open.":[81],"The":[82],"tangent":[84],"differs":[85],"from":[86],"matched":[88],"fixed-exponent":[89],"isomonodromic":[90],"direction,":[91],"but":[92],"admits":[93],"moving-exponent":[96],"curve":[97],"lift.":[98],"Separately,":[99],"reciprocal-head":[101],"canopy":[102],"certificate":[103],"combined":[104],"published":[106],"finite-RH":[107],"rectangular":[109],"time-barrier":[110],"verifications":[111],"bounds":[112],"de":[114],"Bruijn--Newman":[115],"constant":[116],"by":[117],"$\u039b\\le433/2500=0.1732$":[118],"without":[119],"QRH":[121],"assumption.":[122],"These":[123],"global":[127],"analytic":[128],"results":[129],"share":[130],"phase--transverse":[132],"bridge;":[133],"no":[134],"equivalence":[135],"between":[136],"RH":[140],"is":[141],"asserted.":[142]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-10-27T00:00:00"}