{"id":"https://openalex.org/W2155775981","doi":"https://doi.org/10.1090/s0025-5718-2011-02565-2","title":"Gram lines and the average of the real part of the Riemann zeta function","display_name":"Gram lines and the average of the real part of the Riemann zeta function","publication_year":2011,"publication_date":"2011-12-07","ids":{"openalex":"https://openalex.org/W2155775981","doi":"https://doi.org/10.1090/s0025-5718-2011-02565-2","mag":"2155775981"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2011-02565-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02565-2","pdf_url":"https://www.ams.org/mcom/2012-81-279/S0025-5718-2011-02565-2/S0025-5718-2011-02565-2.pdf","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2012-81-279/S0025-5718-2011-02565-2/S0025-5718-2011-02565-2.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014422330","display_name":"Kevin Broughan","orcid":null},"institutions":[{"id":"https://openalex.org/I52179390","display_name":"University of Waikato","ror":"https://ror.org/013fsnh78","country_code":"NZ","type":"education","lineage":["https://openalex.org/I52179390"]}],"countries":["NZ"],"is_corresponding":false,"raw_author_name":"Kevin A. Broughan","raw_affiliation_strings":["University of Waikato, Hamilton, New Zealand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waikato, Hamilton, New Zealand","institution_ids":["https://openalex.org/I52179390"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030296707","display_name":"A.R. Barnett","orcid":null},"institutions":[{"id":"https://openalex.org/I52179390","display_name":"University of Waikato","ror":"https://ror.org/013fsnh78","country_code":"NZ","type":"education","lineage":["https://openalex.org/I52179390"]}],"countries":["NZ"],"is_corresponding":false,"raw_author_name":"A. Ross Barnett","raw_affiliation_strings":["University of Waikato, Hamilton, New Zealand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waikato, Hamilton, New Zealand","institution_ids":["https://openalex.org/I52179390"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I52179390"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.222735,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"81","issue":"279","first_page":"1669","last_page":"1679"},"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/T12812","display_name":"Meromorphic and Entire Functions","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.9984999895095825,"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/mathematics","display_name":"Mathematics","score":0.8689037561416626},{"id":"https://openalex.org/keywords/riemann-zeta-function","display_name":"Riemann zeta function","score":0.7435413002967834},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.5602587461471558},{"id":"https://openalex.org/keywords/critical-line","display_name":"Critical line","score":0.5534465909004211},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.47367674112319946},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.4692910611629486},{"id":"https://openalex.org/keywords/prime-zeta-function","display_name":"Prime zeta function","score":0.46586087346076965},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4606897830963135},{"id":"https://openalex.org/keywords/gram","display_name":"Gram","score":0.43036431074142456},{"id":"https://openalex.org/keywords/arithmetic-zeta-function","display_name":"Arithmetic zeta function","score":0.423788458108902},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.42251184582710266},{"id":"https://openalex.org/keywords/complex-plane","display_name":"Complex plane","score":0.4102933406829834},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.315096378326416},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09094816446304321},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06674164533615112}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8689037561416626},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.7435413002967834},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.5602587461471558},{"id":"https://openalex.org/C2776605787","wikidata":"https://www.wikidata.org/wiki/Q5186739","display_name":"Critical line","level":2,"score":0.5534465909004211},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.47367674112319946},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.4692910611629486},{"id":"https://openalex.org/C71897612","wikidata":"https://www.wikidata.org/wiki/Q1896470","display_name":"Prime zeta function","level":4,"score":0.46586087346076965},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4606897830963135},{"id":"https://openalex.org/C161369605","wikidata":"https://www.wikidata.org/wiki/Q41803","display_name":"Gram","level":3,"score":0.43036431074142456},{"id":"https://openalex.org/C35861355","wikidata":"https://www.wikidata.org/wiki/Q4791135","display_name":"Arithmetic zeta function","level":3,"score":0.423788458108902},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.42251184582710266},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.4102933406829834},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.315096378326416},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09094816446304321},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06674164533615112},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C523546767","wikidata":"https://www.wikidata.org/wiki/Q10876","display_name":"Bacteria","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/s0025-5718-2011-02565-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02565-2","pdf_url":"https://www.ams.org/mcom/2012-81-279/S0025-5718-2011-02565-2/S0025-5718-2011-02565-2.pdf","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:researchcommons.waikato.ac.nz:10289/6183","is_oa":true,"landing_page_url":"https://hdl.handle.net/10289/6183","pdf_url":"https://researchcommons.waikato.ac.nz/bitstreams/6141c9f7-e232-42bd-9304-38e4a5cc31a0/download","source":{"id":"https://openalex.org/S4306400944","display_name":"Research Commons (University of Waikato)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I52179390","host_organization_name":"University of Waikato","host_organization_lineage":["https://openalex.org/I52179390"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.352.6290","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.352.6290","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/journals/mcom/2012-81-279/S0025-5718-2011-02565-2/S0025-5718-2011-02565-2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.583.7873","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.583.7873","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.waikato.ac.nz/~kab/papers/zeta3d.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2011-02565-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02565-2","pdf_url":"https://www.ams.org/mcom/2012-81-279/S0025-5718-2011-02565-2/S0025-5718-2011-02565-2.pdf","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320311746","display_name":"University of Waikato","ror":"https://ror.org/013fsnh78"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2155775981.pdf","grobid_xml":"https://content.openalex.org/works/W2155775981.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W116960172","https://openalex.org/W586135738","https://openalex.org/W1538733621","https://openalex.org/W1980116773","https://openalex.org/W1982849238","https://openalex.org/W2000584812","https://openalex.org/W2072247311","https://openalex.org/W2082225085","https://openalex.org/W2082361850","https://openalex.org/W2106226212","https://openalex.org/W2116781013","https://openalex.org/W2136871397","https://openalex.org/W2314188154","https://openalex.org/W4239066287","https://openalex.org/W4241271931","https://openalex.org/W4245315296","https://openalex.org/W4249328680"],"related_works":["https://openalex.org/W2352880824","https://openalex.org/W4311954729","https://openalex.org/W1486497000","https://openalex.org/W2930994739","https://openalex.org/W2025414628","https://openalex.org/W2165001617","https://openalex.org/W3133260962","https://openalex.org/W4245315296","https://openalex.org/W2504462208","https://openalex.org/W2952563883"],"abstract_inverted_index":{"The":[0],"contours":[1],"$\\Im":[2],"\\Lambda":[3],"(s)=0$":[4],"of":[5,49,60,66],"the":[6,15,29,46,64,70,74,78],"function":[7],"which":[8,20,27,55],"satisfies":[9,52],"$\\zeta":[10,39,50,67],"(1-s)=\\Lambda":[11],"(s)\\zeta":[12],"(s)$":[13,40,51,68],"cross":[14],"critical":[16,30],"strip":[17],"on":[18],"lines":[19],"are":[21],"almost":[22],"horizontal":[23],"and":[24,26,36],"straight,":[25],"cut":[28],"line":[31],"alternately":[32],"at":[33],"Gram":[34,71],"points":[35,37,72],"where":[38],"is":[41],"imaginary.":[42],"When":[43],"suitably":[44],"averaged":[45],"real":[47],"part":[48],"a":[53,58],"relation":[54],"greatly":[56],"extends":[57],"theorem":[59],"Titchmarsh,":[61],"(namely":[62],"that":[63],"average":[65],"over":[69],"has":[73],"value":[75],"2),":[76],"to":[77],"open":[79],"right-hand":[80],"half":[81],"plane":[82],"$\\sigma":[83],">0$.":[84]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-07-02T09:51:11.867554","created_date":"2025-10-10T00:00:00"}