{"id":"https://openalex.org/W7225733258","doi":"https://doi.org/10.5281/zenodo.24099758","title":"Wieferich co-occurrence along exponential sequences: a weighted tail estimate and its application to (star')","display_name":"Wieferich co-occurrence along exponential sequences: a weighted tail estimate and its application to (star')","publication_year":2026,"publication_date":"2026-11-10","ids":{"openalex":"https://openalex.org/W7225733258","doi":"https://doi.org/10.5281/zenodo.24099758"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24099758","doi":"https://doi.org/10.5281/zenodo.24099758","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24099758","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","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24099758","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153107251","display_name":"Chunping Zeng","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zeng, Chunping","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153107251"],"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":0.9911999702453613,"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":0.9911999702453613,"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":0.006500000134110451,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.00019999999494757503,"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"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8583999872207642},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6284999847412109},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.5620999932289124},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4918000102043152},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.47859999537467957},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.4442000091075897},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.43529999256134033},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.42260000109672546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40549999475479126},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.3928999900817871},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.374099999666214},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C64999384","wikidata":"https://www.wikidata.org/wiki/Q3507967","display_name":"Symmetrization","level":2,"score":0.323199987411499},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.29989999532699585},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.2928999960422516},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.2919999957084656},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C38803922","wikidata":"https://www.wikidata.org/wiki/Q95745348","display_name":"Additive function","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.2851000130176544},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.28139999508857727},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C91229774","wikidata":"https://www.wikidata.org/wiki/Q193885","display_name":"Infinitesimal","level":2,"score":0.2775999903678894},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.2583000063896179},{"id":"https://openalex.org/C157351203","wikidata":"https://www.wikidata.org/wiki/Q216906","display_name":"Multiplicative inverse","level":3,"score":0.2500999867916107}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24099758","doi":"https://doi.org/10.5281/zenodo.24099758","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24099758","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.24099759","doi":"https://doi.org/10.5281/zenodo.24099759","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24099759","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.24099758","doi":"https://doi.org/10.5281/zenodo.24099758","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24099758","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Fix":[0],"an":[1,227],"integer":[2],"r":[3],">=":[4],"2":[5,46,137],"and":[6,90,127,161,206,211,224,247,340,371,392],"write":[7],"b_e":[8],"=":[9,45,68,88,116,133,135,180,184,191,362],"r^e":[10,99],"-":[11,100,125],"a'":[12,14],"for":[13,145,287],"<=":[15,186],"A":[16,20,238],"coprime":[17],"to":[18,33,56,304,319],"r.":[19],"recent":[21],"preprint":[22],"(hexagon:2610.00630,":[23],"Zeng)":[24],"reduces":[25],"the":[26,43,48,52,93,108,113,118,167,175,193,207,212,253,263,271,281,288,311,315,320,358,385],"abc":[27],"conjecture,":[28],"within":[29],"a":[30,34,83,221,329],"lattice-theoretic":[31],"framework,":[32],"single":[35],"remaining":[36],"hypothesis":[37,241,298,318],"(star').":[38],"We":[39,324,381],"show":[40],"that":[41],"on":[42,51],"k":[44],"slice":[47],"density-zero":[49],"requirement":[50],"exceptional":[53],"set":[54],"attached":[55],"(star')":[57],"follows":[58],"from":[59],"one":[60],"summability":[61],"condition":[62,195],"K_{r,a'}":[63,67],"<":[64,143,197,245,354],"infinity,":[65],"where":[66],"sum":[69],"over":[70],"q":[71,77,139],"in":[72,252],"C_{r,a'}":[73,81],"of":[74,96,166,174,278],"s_q":[75],"log":[76,138,154,155,349],"(1/D_3(q)":[78],"+":[79,153,364,366],"1/e_0(q)),":[80],"is":[82,92,105,178,257,274,300,327],"three-filter":[84],"congruence":[85],"set,":[86],"D_3(q)":[87],"ord_{q^3}(r),":[89],"e_0(q)":[91],"least":[94,289],"solution":[95],"q^3":[97],"|":[98,123],"a'.":[101],"The":[102,171,296],"main":[103],"result":[104],"unconditional:":[106],"splitting":[107],"governing":[109],"weight":[110],"Xi_r":[111],"by":[112,226,235,302],"block":[114],"d":[115,190],"ord_q(r),":[117],"mass":[119],"constraint":[120],"(prod":[121],"q)^2":[122],"r^d":[124],"1":[126],"Stieltjes":[128],"partial":[129],"summation":[130],"give":[131],"Xi_r^{(2)}":[132],"sum_{s_q":[134],"2}":[136],"/":[140,189,351],"(d_q":[141],"q)":[142],"infinity":[144,198,246,286,355],"every":[146],"base":[147],"r,":[148],"with":[149,259],"explicit":[150],"bound":[151],"O(1":[152],"r);":[156],"no":[157,159,162,383],"abc,":[158],"GRH,":[160],"count":[163],"or":[164],"density":[165],"Wieferich":[168],"primes":[169],"W_r.":[170],"exact":[172,228,359,386],"threshold":[173],"first-moment":[176],"route":[177],"Phi_r(E)":[179],"o(E),":[181],"equivalently":[182,280],"A_r(t)":[183],"sum_{d":[185],"t}":[187],"M_d":[188],"o(t);":[192],"deep-layer":[194],"Xi_r^{(3)}":[196,244],"sits":[199],"strictly":[200,334],"between":[201,335],"Murata's":[202,341],"finiteness":[203],"conjecture":[204,339],"P_{6a}":[205,248],"deep":[208],"reciprocal":[209],"sum,":[210],"barrier":[213,218],"analysis":[214],"(one":[215],"claimed":[216],"radical":[217],"withdrawn":[219],"as":[220],"bookkeeping":[222],"artifact":[223],"replaced":[225],"identity)":[229],"shows":[230],"it":[231,333],"cannot":[232],"be":[233],"reached":[234],"block/radical/partial-summation":[236],"methods.":[237],"Katz":[239],"double-zero":[240],"(D4)":[242,256],"yields":[243],"almost":[249],"surely":[250],"(Borel-Cantelli);":[251],"Haar":[254],"ensemble":[255],"free":[258],"constant":[260],"1,":[261],"while":[262,310],"diagonal":[264],"embedding":[265],"iota(Z)":[266],"has":[267],"measure":[268],"zero,":[269],"so":[270],"missing":[272],"input":[273],"precisely":[275],"pointwise":[276],"rigidity":[277],"iota(r),":[279],"single-variable":[282],"statement":[283],"m(q)":[284,293],"->":[285],"nontrivial":[290],"Teichmuller":[291,321],"representative":[292],"modulo":[294],"q^3.":[295],"bad-set":[297],"(H-j)":[299,326,393],"supported":[301],"computation":[303],"10^6":[305],"(five":[306],"hits,":[307],"zero":[308],"growth),":[309],"same":[312],"data":[313],"correct":[314],"twin-Wieferich":[316],"null":[317],"rate":[322],"(q-1)/q^2.":[323],"prove":[325,382],"not":[328],"GRH":[330],"statement,":[331],"locate":[332],"Katz's":[336],"2015":[337],"equidistribution":[338],"1991":[342],"conjectures,":[343],"record":[344],"ancillary":[345],"unconditional":[346],"results":[347],"(sum_p":[348],"p":[350],"(d_p":[352],"(p-1))":[353],"via":[356],"Erdos-Murty;":[357],"identity":[360],"C_r":[361],"S_W":[363],"S_ord":[365],"LTE_2;":[367],"strict":[368],"one-way":[369],"comparisons),":[370],"specify":[372],"what":[373],"any":[374],"new":[375],"tool":[376],"must":[377],"look":[378],"like":[379],"(S1-S7).":[380],"conjecture:":[384],"threshold,":[387],"Xi_r^{(3)},":[388],"P_{6a},":[389],"P_4,":[390],"P_1":[391],"remain":[394],"open.":[395]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-11T00:00:00"}
