{"id":"https://openalex.org/W7214023586","doi":"https://doi.org/10.48550/arxiv.2609.24248","title":"Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights","display_name":"Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214023586","doi":"https://doi.org/10.48550/arxiv.2609.24248"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24248","pdf_url":"https://arxiv.org/pdf/2609.24248","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/2609.24248","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140614239","display_name":"Yixiu Xiao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiao, Yixiu","raw_affiliation_strings":["School of Mathematical Sciences , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152463983","display_name":"Hongze Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li, Hongze","raw_affiliation_strings":["School of Mathematical Sciences , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , China","institution_ids":["https://openalex.org/I183067930"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I183067930"],"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":1.0,"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":1.0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0,"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"}},{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/kloosterman-sums","display_name":"Kloosterman sums","score":0.9739999771118164},{"id":"https://openalex.org/keywords/sign-changes","display_name":"sign changes","score":0.9309999942779541},{"id":"https://openalex.org/keywords/divisor-sums","display_name":"divisor sums","score":0.8849999904632568},{"id":"https://openalex.org/keywords/uniform-asymptotics","display_name":"uniform asymptotics","score":0.8500000238418579},{"id":"https://openalex.org/keywords/inverse-laplace-transform","display_name":"inverse Laplace transform","score":0.968999981880188}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9093000292778015},{"id":"https://openalex.org/C2778217198","wikidata":"https://www.wikidata.org/wiki/Q556977","display_name":"Cutoff","level":2,"score":0.6211000084877014},{"id":"https://openalex.org/C34282223","wikidata":"https://www.wikidata.org/wiki/Q6421354","display_name":"Kloosterman sum","level":2,"score":0.6018999814987183},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.51419997215271},{"id":"https://openalex.org/C24209939","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic formula","level":2,"score":0.5138999819755554},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.5070000290870667},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4528999924659729},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.4449000060558319},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.44130000472068787},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4381999969482422},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.40119999647140503},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3873000144958496},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3813000023365021},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.38029998540878296},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.3788999915122986},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3409000039100647},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C17732976","wikidata":"https://www.wikidata.org/wiki/Q7511782","display_name":"Sieve (category theory)","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C141513077","wikidata":"https://www.wikidata.org/wiki/Q378542","display_name":"Independent and identically distributed random variables","level":3,"score":0.302700012922287},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.29429998993873596},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.27639999985694885},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.26829999685287476},{"id":"https://openalex.org/C30014739","wikidata":"https://www.wikidata.org/wiki/Q5159445","display_name":"Cone (formal languages)","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.2578999996185303},{"id":"https://openalex.org/C106347477","wikidata":"https://www.wikidata.org/wiki/Q5384228","display_name":"Equating","level":3,"score":0.2522999942302704}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24248","pdf_url":"https://arxiv.org/pdf/2609.24248","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.2609.24248","doi":"https://doi.org/10.48550/arxiv.2609.24248","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24248","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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.24248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24248","pdf_url":"https://arxiv.org/pdf/2609.24248","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/G7895294006","display_name":null,"funder_award_id":"12171311","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320320965","display_name":"University of New South Wales","ror":"https://ror.org/03r8z3t63"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322725","display_name":"China Scholarship Council","ror":"https://ror.org/04atp4p48"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214023586.pdf","grobid_xml":"https://content.openalex.org/works/W7214023586.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2291317981","https://openalex.org/W3044633612"],"related_works":["https://openalex.org/W7213996050","https://openalex.org/W4404648977","https://openalex.org/W2592459126","https://openalex.org/W2224288049","https://openalex.org/W7168239861","https://openalex.org/W4395053196","https://openalex.org/W2962722575","https://openalex.org/W258201547","https://openalex.org/W1620496769","https://openalex.org/W7026696319"],"abstract_inverted_index":{"We":[0,49],"prove":[1],"an":[2,35],"asymptotic":[3],"formula":[4],"for":[5,24],"correlations":[6],"of":[7,45,86],"divisor":[8],"sums":[9],"with":[10],"quadratic":[11],"cutoff":[12,47,89],"functions":[13],"shifted":[14],"independently":[15],"by":[16,52,67],"\\(s,u\\in[0,1/2]\\).":[17],"The":[18,31,70],"error":[19],"is":[20,34,94],"\\(O_g(Y/(\\log":[21],"Y)^2)\\),":[22],"uniformly":[23],"\\(Y^{1/5}\\leq":[25],"R\\leq":[26],"Y^{1/3}\\),":[27],"including":[28,83],"coincident":[29],"shifts.":[30],"main":[32],"term":[33],"explicit":[36],"bilinear":[37],"form":[38],"in":[39,56,96],"the":[40,46,54,57,63,74,80,84,87,91],"first":[41],"and":[42,61,90],"second":[43],"derivatives":[44],"functions.":[48],"obtain":[50],"it":[51],"taking":[53],"residue":[55],"outer":[58],"Mellin":[59],"variable":[60],"evaluating":[62],"remaining":[64,92],"double":[65],"integral":[66],"Laplace":[68],"inversion.":[69],"present":[71],"paper":[72],"establishes":[73],"divisor-sum":[75],"estimate.":[76],"Its":[77],"application":[78],"to":[79],"sign-change":[81],"problem,":[82],"choice":[85],"prime":[88],"parameters,":[93],"treated":[95],"Part":[97],"II.":[98]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}