{"id":"https://openalex.org/W2592326766","doi":"https://doi.org/10.1090/s0025-5718-2011-02452-x","title":"An amortized-complexity method to compute the Riemann zeta function","display_name":"An amortized-complexity method to compute the Riemann zeta function","publication_year":2011,"publication_date":"2011-01-25","ids":{"openalex":"https://openalex.org/W2592326766","doi":"https://doi.org/10.1090/s0025-5718-2011-02452-x","mag":"2592326766"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2011-02452-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02452-x","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2011-02452-X/S0025-5718-2011-02452-X.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/2011-80-275/S0025-5718-2011-02452-X/S0025-5718-2011-02452-X.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034913308","display_name":"Ghaith A. Hiary","orcid":"https://orcid.org/0000-0002-4517-3541"},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]},{"id":"https://openalex.org/I40036882","display_name":"Institute for Advanced Study","ror":"https://ror.org/00f809463","country_code":"US","type":"facility","lineage":["https://openalex.org/I40036882"]}],"countries":["CA","US"],"is_corresponding":true,"raw_author_name":"Ghaith Hiary","raw_affiliation_strings":["Institute for advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540","University of Waterloo, Department of Pure Mathematics, 200 University Avenue W, Waterloo, ON N2L 3G1, Canada"],"affiliations":[{"raw_affiliation_string":"Institute for advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540","institution_ids":["https://openalex.org/I40036882"]},{"raw_affiliation_string":"University of Waterloo, Department of Pure Mathematics, 200 University Avenue W, Waterloo, ON N2L 3G1, Canada","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5034913308"],"corresponding_institution_ids":["https://openalex.org/I151746483","https://openalex.org/I40036882"],"apc_list":null,"apc_paid":null,"fwci":2.6344,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.89604547,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"80","issue":"275","first_page":"1785","last_page":"1796"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8390606641769409},{"id":"https://openalex.org/keywords/riemann-zeta-function","display_name":"Riemann zeta function","score":0.6169919371604919},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.5294399261474609},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5187475681304932},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47766029834747314},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.47399795055389404},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.46192309260368347},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4536043405532837},{"id":"https://openalex.org/keywords/amortized-analysis","display_name":"Amortized analysis","score":0.43966278433799744},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3817332983016968},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36519333720207214},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2556740343570709},{"id":"https://openalex.org/keywords/data-structure","display_name":"Data structure","score":0.17313796281814575},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10652366280555725}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8390606641769409},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.6169919371604919},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.5294399261474609},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5187475681304932},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47766029834747314},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47399795055389404},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.46192309260368347},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4536043405532837},{"id":"https://openalex.org/C142417499","wikidata":"https://www.wikidata.org/wiki/Q331716","display_name":"Amortized analysis","level":3,"score":0.43966278433799744},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3817332983016968},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36519333720207214},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2556740343570709},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.17313796281814575},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10652366280555725},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-2011-02452-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02452-x","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2011-02452-X/S0025-5718-2011-02452-X.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:research-information.bris.ac.uk:openaire_cris_publications/84e556f8-e3f1-466f-938c-5feac6b48fe5","is_oa":false,"landing_page_url":"https://research-information.bris.ac.uk/en/publications/84e556f8-e3f1-466f-938c-5feac6b48fe5","pdf_url":null,"source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Hiary, G 2011, 'An amortized-complexity method to compute the Riemann zeta function', Mathematics of Computation, vol. 80, no. 275, pp. 1785-1796. https://doi.org/10.1090/S0025-5718-2011-02452-X","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.769.3361","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.769.3361","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1002.2443.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2011-02452-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2011-02452-x","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2011-02452-X/S0025-5718-2011-02452-X.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":[{"id":"https://openalex.org/G1207306512","display_name":null,"funder_award_id":"DMS-0635607","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2182150653","display_name":"FRG: L-functions and Modular Forms","funder_award_id":"0757627","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2592326766.pdf","grobid_xml":"https://content.openalex.org/works/W2592326766.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1634565696","https://openalex.org/W2043906885","https://openalex.org/W2082478941","https://openalex.org/W3103878215","https://openalex.org/W4251477635"],"related_works":["https://openalex.org/W2610184042","https://openalex.org/W2964407532","https://openalex.org/W4294651012","https://openalex.org/W1513480133","https://openalex.org/W2986559382","https://openalex.org/W2086661656","https://openalex.org/W3200545397","https://openalex.org/W4288019650","https://openalex.org/W2909634389","https://openalex.org/W3129435903"],"abstract_inverted_index":{"A":[0],"practical":[1],"method":[2,12,212,289],"to":[3,263,286],"compute":[4,14],"the":[5,200,204,211,217,257,269],"Riemann":[6],"zeta":[7],"function":[8],"is":[9,199],"presented.":[10],"The":[11,288],"can":[13],"<inline-formula":[15,55,104,156,225,305],"content-type=\"math/mathml\">":[16,56,105,157,226,306],"<mml:math":[17,57,106,158,227,307],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[18,58,107,159,228,308],"alttext=\"zeta":[19],"left-parenthesis":[20,168,317],"1":[21,64,118,163,169,232,312,318],"slash":[22,65,119,164,233,313],"2":[23],"plus":[24,114,166,315],"i":[25],"t":[26],"right-parenthesis\">":[27,170,319],"<mml:semantics>":[28,70,123,171,235,320],"<mml:mrow>":[29,71,124],"<mml:mi>":[30],"\u03b6":[31],"</mml:mi>":[32],"<mml:mo":[33,44,72,89,125,143,184,187,333,336],"stretchy=\"false\">(</mml:mo>":[34,185,334],"<mml:mn>1</mml:mn>":[35,81,135,176,186,240,325,335],"<mml:mrow":[36,79,82,133,136,174,177,238,241,323,326],"class=\"MJX-TeXAtom-ORD\">":[37,80,83,134,137,175,178,239,242,324,327],"<mml:mo>/</mml:mo>":[38,84,138,179,243,328],"</mml:mrow>":[39,46,85,87,94,139,141,145,180,189,244,246,329,338],"<mml:mn>2</mml:mn>":[40],"<mml:mo>+</mml:mo>":[41,130,182,331],"<mml:mi>i</mml:mi>":[42],"<mml:mi>t</mml:mi>":[43],"stretchy=\"false\">)</mml:mo>":[45,188,337],"<mml:annotation":[47,95,146,191,248,340],"encoding=\"application/x-tex\">\\zeta":[48],"(1/2+it)</mml:annotation>":[49],"</mml:semantics>":[50,99,148,193,250,342],"</mml:math>":[51,100,149,194,251,343],"</inline-formula>":[52,101,150,195,252,344],"at":[53],"any":[54],"alttext=\"left":[59],"floor":[60],"upper":[61,109,112,115],"T":[62,110,113,116,161,230,310],"Superscript":[63,117,162,231,311],"4":[66,120,165,314],"Baseline":[67,121],"right":[68],"floor\">":[69],"fence=\"false\"":[73,90],"stretchy=\"false\">":[74,91],"\u230a":[75],"</mml:mo>":[76,93],"<mml:msup>":[77,131,172,236,321],"<mml:mi>T</mml:mi>":[78,127,129,132,173,237,322],"<mml:mn>4</mml:mn>":[86,140,181,245,330],"</mml:msup>":[88,142,190,247,339],"\u230b":[92],"encoding=\"application/x-tex\">\\lfloor":[96],"T^{1/4}":[97],"\\rfloor</mml:annotation>":[98],"points":[102],"in":[103],"alttext=\"left-bracket":[108],"comma":[111],"right-bracket\">":[122],"stretchy=\"false\">[</mml:mo>":[126],"<mml:mo>,</mml:mo>":[128],"stretchy=\"false\">]</mml:mo>":[144],"encoding=\"application/x-tex\">[T,T+T^{1/4}]</mml:annotation>":[147],"using":[151],"an":[152],"<italic>average</italic>":[153],"time":[154],"of":[155,259,277,295,304],"alttext=\"upper":[160,229,309],"o":[167,316],"<mml:mi>o</mml:mi>":[183,332],"encoding=\"application/x-tex\">T^{1/4+o(1)}</mml:annotation>":[192,341],"per":[196],"point.":[197],"This":[198],"same":[201],"complexity":[202,303],"as":[203],"Odlyzko-Sch\u00f6nhage":[205,218],"algorithm":[206,219],"over":[207,220],"that":[208],"interval.":[209],"Although":[210],"far":[213],"from":[214],"competes":[215],"with":[216,299],"intervals":[221],"much":[222],"longer":[223],"than":[224],"4\">":[234],"encoding=\"application/x-tex\">T^{1/4}</mml:annotation>":[249],",":[253],"it":[254,265],"still":[255],"has":[256,290],"advantages":[258],"being":[260],"elementary,":[261],"simple":[262],"implement,":[264],"does":[266],"not":[267],"use":[268],"fast":[270],"Fourier":[271],"transform":[272],"or":[273],"require":[274],"large":[275],"amounts":[276],"storage":[278],"space,":[279],"and":[280,293],"its":[281,300],"error":[282],"terms":[283],"are":[284],"easy":[285],"control.":[287],"been":[291],"implemented,":[292],"results":[294],"timing":[296],"experiments":[297],"agree":[298],"theoretical":[301],"amortized":[302],".":[345]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
