{"id":"https://openalex.org/W3023740157","doi":"https://doi.org/10.1090/mcom/3599","title":"Counting zeros of Dirichlet $L$-functions","display_name":"Counting zeros of Dirichlet $L$-functions","publication_year":2020,"publication_date":"2020-10-21","ids":{"openalex":"https://openalex.org/W3023740157","doi":"https://doi.org/10.1090/mcom/3599","mag":"3023740157"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3599","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3599","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2005.02989","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102805894","display_name":"Michael A. Bennett","orcid":"https://orcid.org/0000-0002-8888-6218"},"institutions":[{"id":"https://openalex.org/I141945490","display_name":"University of British Columbia","ror":"https://ror.org/03rmrcq20","country_code":"CA","type":"education","lineage":["https://openalex.org/I141945490"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Michael A. Bennett","raw_affiliation_strings":["Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","institution_ids":["https://openalex.org/I141945490"]},{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada","institution_ids":["https://openalex.org/I141945490"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087476929","display_name":"Greg Martin","orcid":"https://orcid.org/0000-0002-8476-9495"},"institutions":[{"id":"https://openalex.org/I141945490","display_name":"University of British Columbia","ror":"https://ror.org/03rmrcq20","country_code":"CA","type":"education","lineage":["https://openalex.org/I141945490"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Greg Martin","raw_affiliation_strings":["Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","institution_ids":["https://openalex.org/I141945490"]},{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada","institution_ids":["https://openalex.org/I141945490"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075461741","display_name":"Kevin O\u2019Bryant","orcid":null},"institutions":[{"id":"https://openalex.org/I121847817","display_name":"The Graduate Center, CUNY","ror":"https://ror.org/00awd9g61","country_code":"US","type":"education","lineage":["https://openalex.org/I121847817"]},{"id":"https://openalex.org/I142393192","display_name":"College of Staten Island","ror":"https://ror.org/02p179j44","country_code":"US","type":"education","lineage":["https://openalex.org/I142393192"]},{"id":"https://openalex.org/I174216632","display_name":"City University of New York","ror":"https://ror.org/00453a208","country_code":"US","type":"education","lineage":["https://openalex.org/I174216632"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kevin O\u2019Bryant","raw_affiliation_strings":["Department of Mathematics, City University of New York, College of Staten Island and The Graduate Center, 2800 Victory Boulevard, Staten Island, New York 10314"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, City University of New York, College of Staten Island and The Graduate Center, 2800 Victory Boulevard, Staten Island, New York 10314","institution_ids":["https://openalex.org/I142393192","https://openalex.org/I174216632","https://openalex.org/I121847817"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043988338","display_name":"Andrew Rechnitzer","orcid":"https://orcid.org/0000-0002-4386-3207"},"institutions":[{"id":"https://openalex.org/I141945490","display_name":"University of British Columbia","ror":"https://ror.org/03rmrcq20","country_code":"CA","type":"education","lineage":["https://openalex.org/I141945490"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Andrew Rechnitzer","raw_affiliation_strings":["Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada","institution_ids":["https://openalex.org/I141945490"]},{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada","institution_ids":["https://openalex.org/I141945490"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5021,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.61371337,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"90","issue":"329","first_page":"1455","last_page":"1482"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9898999929428101,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9769999980926514,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6755934357643127},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5516563057899475},{"id":"https://openalex.org/keywords/pi","display_name":"Pi","score":0.5221602916717529},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.5006778240203857},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4801768362522125},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.46614375710487366},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3374463617801666},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22909483313560486},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.105130136013031}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6755934357643127},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5516563057899475},{"id":"https://openalex.org/C53009064","wikidata":"https://www.wikidata.org/wiki/Q167","display_name":"Pi","level":2,"score":0.5221602916717529},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.5006778240203857},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4801768362522125},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.46614375710487366},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3374463617801666},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22909483313560486},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.105130136013031},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/mcom/3599","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3599","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2005.02989","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2005.02989","pdf_url":"https://arxiv.org/pdf/2005.02989","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},{"id":"mag:3023740157","is_oa":true,"landing_page_url":"http://arxiv.org/pdf/2005.02989.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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},{"id":"doi:10.48550/arxiv.2005.02989","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2005.02989","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2005.02989","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2005.02989","pdf_url":"https://arxiv.org/pdf/2005.02989","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4841995757","display_name":null,"funder_award_id":"63146-00 51","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W164887593","https://openalex.org/W1549756090","https://openalex.org/W1556596029","https://openalex.org/W1979499160","https://openalex.org/W1989563529","https://openalex.org/W1999885188","https://openalex.org/W2024731156","https://openalex.org/W2036605016","https://openalex.org/W2963987338","https://openalex.org/W3101695955"],"related_works":["https://openalex.org/W3093614117","https://openalex.org/W3184499602","https://openalex.org/W1969932567","https://openalex.org/W3136910610","https://openalex.org/W2293670472","https://openalex.org/W2974839816","https://openalex.org/W2136023802","https://openalex.org/W2337795388","https://openalex.org/W3101320175","https://openalex.org/W2074108977","https://openalex.org/W2949366143","https://openalex.org/W2949858172","https://openalex.org/W2593126346","https://openalex.org/W3167244677","https://openalex.org/W3021336475","https://openalex.org/W2097496186","https://openalex.org/W2041305764","https://openalex.org/W2951465737","https://openalex.org/W2112607951"],"abstract_inverted_index":{"We":[0,76],"give":[1,77],"explicit":[2],"upper":[3],"and":[4,21,32,84],"lower":[5],"bounds":[6],"for":[7,81,97],"$N(T,\\chi":[8],")$,":[9],"the":[10,87],"number":[11],"of":[12,14],"zeros":[13],"a":[15,91],"Dirichlet":[16],"$L$-function":[17],"with":[18],"character":[19],"$\\chi$":[20,28],"height":[22],"at":[23],"most":[24],"$T$.":[25,85],"Suppose":[26],"that":[27,33],"has":[29],"conductor":[30],"$q>1$,":[31],"$T\\geq":[34],"5/7$.":[35],"If":[36],"$\\ell":[37],"=\\log":[38],"\\frac":[39,52,56],"{q(T+2)}{2\\pi":[40],"}>":[41],"1.567$,":[42],"then":[43],"\\begin{equation*}":[44],"\\left":[45,50],"|":[46,64],"N(T,\\chi":[47],")":[48,62,72],"-":[49,73],"(":[51],"{T}{\\pi":[53],"}":[54],"\\log":[55,70],"{qT}{2\\pi":[57],"e}":[58],"-\\frac":[59],"{\\chi":[60],"(-1)}{4}\\right":[61],"\\right":[63],"\\le":[65],"0.22737":[66],"\\ell":[67],"+":[68],"2":[69],"(1+\\ell":[71],"0.5.":[74],"\\end{equation*}":[75],"slightly":[78],"stronger":[79],"results":[80],"small":[82],"$q$":[83],"Along":[86],"way,":[88],"we":[89],"prove":[90],"new":[92],"bound":[93],"on":[94],"$|L(s,\\chi":[95],")|$":[96],"$\\sigma":[98],"<-1/2$.":[99]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}