{"id":"https://openalex.org/W7224865898","doi":"https://doi.org/10.5281/zenodo.23797605","title":"Quadratic floor sums II: fibre sums and quartic traces at conductor 8\u2113p, formalized in Lean","display_name":"Quadratic floor sums II: fibre sums and quartic traces at conductor 8\u2113p, formalized in Lean","publication_year":2026,"publication_date":"2026-10-30","ids":{"openalex":"https://openalex.org/W7224865898","doi":"https://doi.org/10.5281/zenodo.23797605"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23797605","doi":"https://doi.org/10.5281/zenodo.23797605","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797605","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":"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"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23797605","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158900835","display_name":"Joseph S. Farland","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Farland, Joseph S.","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0007-6809-804X","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5158900835"],"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":0.9345999956130981,"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":0.9345999956130981,"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":0.06199999898672104,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.0006000000284984708,"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/class-number","display_name":"class number","score":0.5569999814033508},{"id":"https://openalex.org/keywords/lean-4-formalization","display_name":"Lean 4 formalization","score":0.6510000228881836},{"id":"https://openalex.org/keywords/mathlib","display_name":"Mathlib","score":0.8840000033378601}],"concepts":[{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.8508999943733215},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8400999903678894},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.6766999959945679},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5835000276565552},{"id":"https://openalex.org/C2780861071","wikidata":"https://www.wikidata.org/wiki/Q1062934","display_name":"Character (mathematics)","level":2,"score":0.5684000253677368},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5383999943733215},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5310999751091003},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.5103999972343445},{"id":"https://openalex.org/C129960964","wikidata":"https://www.wikidata.org/wiki/Q694114","display_name":"Bernoulli number","level":2,"score":0.4345000088214874},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4293000102043152},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42160001397132874},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4196999967098236},{"id":"https://openalex.org/C34800285","wikidata":"https://www.wikidata.org/wiki/Q5159395","display_name":"Conductor","level":2,"score":0.40450000762939453},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.33559998869895935},{"id":"https://openalex.org/C12817185","wikidata":"https://www.wikidata.org/wiki/Q878259","display_name":"Quadratic residue","level":2,"score":0.3262999951839447},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3027999997138977},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3003000020980835},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2720000147819519},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.27160000801086426},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.27090001106262207},{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.2637999951839447},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2615000009536743},{"id":"https://openalex.org/C6026789","wikidata":"https://www.wikidata.org/wiki/Q948010","display_name":"Primitive root modulo n","level":3,"score":0.25839999318122864},{"id":"https://openalex.org/C23990920","wikidata":"https://www.wikidata.org/wiki/Q518831","display_name":"Bernoulli process","level":3,"score":0.2506999969482422}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.23797605","doi":"https://doi.org/10.5281/zenodo.23797605","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797605","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":"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"},{"id":"doi:10.5281/zenodo.23797606","doi":"https://doi.org/10.5281/zenodo.23797606","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797606","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":"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"},{"id":"doi:10.5281/zenodo.23883390","doi":"https://doi.org/10.5281/zenodo.23883390","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23883390","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":"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":"doi:10.5281/zenodo.23797605","doi":"https://doi.org/10.5281/zenodo.23797605","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797605","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":"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"},"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/W7224383369","https://openalex.org/W7223653451","https://openalex.org/W7224736788","https://openalex.org/W7223465783","https://openalex.org/W2024446619","https://openalex.org/W4402831647","https://openalex.org/W4416292189","https://openalex.org/W1552161001","https://openalex.org/W1993515438","https://openalex.org/W4242452568"],"abstract_inverted_index":{"For":[0,101],"a":[1,102,110,124,246,256,277],"positive":[2],"odd":[3,7,48,188],"integer":[4],"A,":[5,80],"an":[6,41,175],"prime":[8,103,189],"p":[9,29,132,201],"not":[10,209,220],"dividing":[11],"A":[12,251],"and":[13,73,109,162,174,222,275],"integers":[14,185],"B,":[15],"C,":[16],"the":[17,64,76,84,89,118,136,150,155,159,167,194,198,205,210,223,227,237,241,249,272],"alternating":[18],"floor":[19,145],"sum":[20,23,85],"S_{A,B,C}(p)":[21],"=":[22,92,99,105,171,202,252],"over":[24,75,88],"0":[25],"<=":[26],"n":[27],"<":[28],"of":[30,43,47,66,86,121,128,144,181,240,248],"(-1)^floor((An^2":[31],"+":[32,34],"Bn":[33],"C)/p)":[35],"is,":[36],"for":[37,123,158,186,197],"admissible":[38],"(B,":[39],"C),":[40],"average":[42],"first":[44],"Bernoulli":[45,138,224],"numbers":[46],"characters":[49],"vartheta_{p,psi}":[50],"modulo":[51,79,114],"8Ap":[52],"weighted":[53],"by":[54,71],"conj(psi(B))^2;":[55],"this":[56,116],"was":[57],"formalized":[58],"earlier.":[59],"Here":[60],"we":[61],"prove":[62,149],"what":[63],"family":[65],"these":[67,266],"sums":[68],"determines.":[69],"Weighted":[70],"psi_0(b)^2":[72],"summed":[74],"units":[77],"b":[78],"it":[81],"gives":[82],"minus":[83],"B_{1,vartheta_{p,psi}}":[87],"fibre":[90],"psi^2":[91],"psi_0^2,":[93],"which":[94,212,244],"is":[95,208,213,226,245],"real":[96,119],"when":[97,131],"psi_0^4":[98],"1.":[100],"l":[104],"1":[106,253],"(mod":[107],"4)":[108],"quartic":[111,125],"character":[112,126],"xi":[113],"l,":[115,134],"separates":[117],"part":[120],"B_{1,Theta_{l,p}},":[122],"Theta_{l,p}":[127],"conductor":[129,195,218],"8lp":[130,207],"!=":[133],"from":[135],"quadratic":[137],"number":[139,225],"B_{1,chi_{-8lp}},":[140],"as":[141],"signed":[142],"combinations":[143],"sums.":[146],"We":[147],"also":[148,235],"Euler-factor":[151],"relation":[152],"that":[153,178,270],"puts":[154],"earlier":[156],"formulas":[157],"coefficients":[160],"three":[161],"five":[163],"in":[164,280],"class-number":[165],"shape,":[166],"quarter-range":[168],"formula":[169],"B_{1,vartheta_{p,psi}}^{(8Ap)}":[170],"-sum_{0