{"id":"https://openalex.org/W4200426778","doi":"https://doi.org/10.3390/sym13122410","title":"The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann\u2019s Zeta-Function","display_name":"The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann\u2019s Zeta-Function","publication_year":2021,"publication_date":"2021-12-13","ids":{"openalex":"https://openalex.org/W4200426778","doi":"https://doi.org/10.3390/sym13122410"},"language":"en","primary_location":{"id":"doi:10.3390/sym13122410","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122410","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2410/pdf?version=1639392739","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/12/2410/pdf?version=1639392739","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082059492","display_name":"Janyarak Tongsomporn","orcid":"https://orcid.org/0000-0001-8744-2298"},"institutions":[{"id":"https://openalex.org/I96916377","display_name":"Walailak University","ror":"https://ror.org/04b69g067","country_code":"TH","type":"education","lineage":["https://openalex.org/I96916377"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Janyarak Tongsomporn","raw_affiliation_strings":["School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand","institution_ids":["https://openalex.org/I96916377"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041496925","display_name":"Saeree Wananiyakul","orcid":"https://orcid.org/0009-0007-9269-2487"},"institutions":[{"id":"https://openalex.org/I96916377","display_name":"Walailak University","ror":"https://ror.org/04b69g067","country_code":"TH","type":"education","lineage":["https://openalex.org/I96916377"]}],"countries":["TH"],"is_corresponding":false,"raw_author_name":"Saeree Wananiyakul","raw_affiliation_strings":["School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand","institution_ids":["https://openalex.org/I96916377"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5109911656","display_name":"J\u00f6rn Steuding","orcid":null},"institutions":[{"id":"https://openalex.org/I25974101","display_name":"University of W\u00fcrzburg","ror":"https://ror.org/00fbnyb24","country_code":"DE","type":"education","lineage":["https://openalex.org/I25974101"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"J\u00f6rn Steuding","raw_affiliation_strings":["Department of Mathematics, W\u00fcrzburg University, Am Hubland, 97218 W\u00fcrzburg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, W\u00fcrzburg University, Am Hubland, 97218 W\u00fcrzburg, Germany","institution_ids":["https://openalex.org/I25974101"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5082059492"],"corresponding_institution_ids":["https://openalex.org/I96916377"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.5161,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.89621318,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"13","issue":"12","first_page":"2410","last_page":"2410"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9986000061035156,"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/T11822","display_name":"Analytic and geometric function theory","score":0.9959999918937683,"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/riemann-zeta-function","display_name":"Riemann zeta function","score":0.8733124732971191},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.7683438062667847},{"id":"https://openalex.org/keywords/particular-values-of-riemann-zeta-function","display_name":"Particular values of Riemann zeta function","score":0.6916342377662659},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6844456791877747},{"id":"https://openalex.org/keywords/critical-line","display_name":"Critical line","score":0.6511622667312622},{"id":"https://openalex.org/keywords/prime-zeta-function","display_name":"Prime zeta function","score":0.5742525458335876},{"id":"https://openalex.org/keywords/explicit-formulae","display_name":"Explicit formulae","score":0.5693586468696594},{"id":"https://openalex.org/keywords/z-function","display_name":"Z function","score":0.5517714023590088},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5207624435424805},{"id":"https://openalex.org/keywords/arithmetic-zeta-function","display_name":"Arithmetic zeta function","score":0.50227952003479},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49449941515922546},{"id":"https://openalex.org/keywords/riemann-xi-function","display_name":"Riemann Xi function","score":0.4760294258594513},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.4517771303653717},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.42402297258377075},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.368074893951416},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19477838277816772},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07004180550575256}],"concepts":[{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.8733124732971191},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.7683438062667847},{"id":"https://openalex.org/C203513110","wikidata":"https://www.wikidata.org/wiki/Q1073118","display_name":"Particular values of Riemann zeta function","level":5,"score":0.6916342377662659},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6844456791877747},{"id":"https://openalex.org/C2776605787","wikidata":"https://www.wikidata.org/wiki/Q5186739","display_name":"Critical line","level":2,"score":0.6511622667312622},{"id":"https://openalex.org/C71897612","wikidata":"https://www.wikidata.org/wiki/Q1896470","display_name":"Prime zeta function","level":4,"score":0.5742525458335876},{"id":"https://openalex.org/C27783845","wikidata":"https://www.wikidata.org/wiki/Q5421272","display_name":"Explicit formulae","level":2,"score":0.5693586468696594},{"id":"https://openalex.org/C111993293","wikidata":"https://www.wikidata.org/wiki/Q8063364","display_name":"Z function","level":4,"score":0.5517714023590088},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5207624435424805},{"id":"https://openalex.org/C35861355","wikidata":"https://www.wikidata.org/wiki/Q4791135","display_name":"Arithmetic zeta function","level":3,"score":0.50227952003479},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49449941515922546},{"id":"https://openalex.org/C48902493","wikidata":"https://www.wikidata.org/wiki/Q2152235","display_name":"Riemann Xi function","level":3,"score":0.4760294258594513},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4517771303653717},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42402297258377075},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.368074893951416},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19477838277816772},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07004180550575256},{"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/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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym13122410","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122410","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2410/pdf?version=1639392739","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:ad69b468df5c45e492786d3c718e40e4","is_oa":true,"landing_page_url":"https://doaj.org/article/ad69b468df5c45e492786d3c718e40e4","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 13, Iss 12, p 2410 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/12/2410/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13122410","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"},{"id":"pmh:oai:opus.bibliothek.uni-wuerzburg.de:25226","is_oa":true,"landing_page_url":"https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-252261","pdf_url":null,"source":{"id":"https://openalex.org/S4306401635","display_name":"Online Publication Service of W\u00fcrzburg University (W\u00fcrzburg University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I25974101","host_organization_name":"University of W\u00fcrzburg","host_organization_lineage":["https://openalex.org/I25974101"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym13122410","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122410","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2410/pdf?version=1639392739","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.4300000071525574,"display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G345598533","display_name":null,"funder_award_id":"563NRCT322800","funder_id":"https://openalex.org/F4320326188","funder_display_name":"Walailak University"},{"id":"https://openalex.org/G7621105261","display_name":null,"funder_award_id":"563NRCT322800","funder_id":"https://openalex.org/F4320322817","funder_display_name":"National Research Council of Thailand"}],"funders":[{"id":"https://openalex.org/F4320322817","display_name":"National Research Council of Thailand","ror":"https://ror.org/018wfhg78"},{"id":"https://openalex.org/F4320326188","display_name":"Walailak University","ror":"https://ror.org/04b69g067"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4200426778.pdf","grobid_xml":"https://content.openalex.org/works/W4200426778.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W829389795","https://openalex.org/W2078065964","https://openalex.org/W2086661656","https://openalex.org/W2506039256","https://openalex.org/W2964090578","https://openalex.org/W4243872063","https://openalex.org/W6724448369"],"related_works":["https://openalex.org/W4377131105","https://openalex.org/W2003633276","https://openalex.org/W2125530360","https://openalex.org/W2352880824","https://openalex.org/W4200426778","https://openalex.org/W2025414628","https://openalex.org/W3035691016","https://openalex.org/W2963554201","https://openalex.org/W41617814","https://openalex.org/W3014201687"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"prove":[4],"an":[5,43],"asymptotic":[6],"formula":[7],"for":[8],"the":[9,12,15,19,23,34,38,46,62],"sum":[10],"of":[11,14,22,45],"values":[13],"periodic":[16],"zeta-function":[17,25],"at":[18],"nontrivial":[20],"zeros":[21],"Riemann":[24,65],"(up":[26],"to":[27,50],"some":[28],"height)":[29],"which":[30],"are":[31],"symmetrical":[32],"on":[33],"real":[35],"line":[36],"and":[37],"critical":[39],"line.":[40],"This":[41],"is":[42],"extension":[44],"previous":[47],"results":[48],"due":[49],"Garunk\u0161tis,":[51],"Kalpokas,":[52],"and,":[53],"more":[54],"recently,":[55],"Sowa.":[56],"Whereas":[57],"Sowa\u2019s":[58],"approach":[59],"was":[60],"assuming":[61],"yet":[63],"unproved":[64],"hypothesis,":[66],"our":[67],"result":[68],"holds":[69],"unconditionally.":[70]},"counts_by_year":[{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2021-12-31T00:00:00"}
