{"id":"https://openalex.org/W2045047334","doi":"https://doi.org/10.1090/s0025-5718-09-02249-2","title":"Computing prime harmonic sums","display_name":"Computing prime harmonic sums","publication_year":2009,"publication_date":"2009-04-03","ids":{"openalex":"https://openalex.org/W2045047334","doi":"https://doi.org/10.1090/s0025-5718-09-02249-2","mag":"2045047334"},"language":"lv","primary_location":{"id":"doi:10.1090/s0025-5718-09-02249-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02249-2","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02249-2/S0025-5718-09-02249-2.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/2009-78-268/S0025-5718-09-02249-2/S0025-5718-09-02249-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057710320","display_name":"Eric Bach","orcid":"https://orcid.org/0000-0001-6200-5462"},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Eric Bach","raw_affiliation_strings":["Computer Sciences Department, University of Wisconsin-Madison, 1210 W. Dayton Street, Madison, Wisconsin 53706"],"affiliations":[{"raw_affiliation_string":"Computer Sciences Department, University of Wisconsin-Madison, 1210 W. Dayton Street, Madison, Wisconsin 53706","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026639653","display_name":"Dominic Klyve","orcid":"https://orcid.org/0000-0002-4424-3130"},"institutions":[{"id":"https://openalex.org/I176902721","display_name":"Carthage College","ror":"https://ror.org/00wkay776","country_code":"US","type":"education","lineage":["https://openalex.org/I176902721"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dominic Klyve","raw_affiliation_strings":["Department of Mathematics, Carthage College, 2001 Alford Drive, Kenosha, Wisconsin 53140"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Carthage College, 2001 Alford Drive, Kenosha, Wisconsin 53140","institution_ids":["https://openalex.org/I176902721"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025020420","display_name":"Jonathan Sorenson","orcid":"https://orcid.org/0000-0002-8887-5957"},"institutions":[{"id":"https://openalex.org/I110678235","display_name":"Butler University","ror":"https://ror.org/05gq3a412","country_code":"US","type":"education","lineage":["https://openalex.org/I110678235"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jonathan Sorenson","raw_affiliation_strings":["Computer Science and Software Engineering, Butler University, Indianapolis, Indiana 46208"],"affiliations":[{"raw_affiliation_string":"Computer Science and Software Engineering, Butler University, Indianapolis, Indiana 46208","institution_ids":["https://openalex.org/I110678235"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5057710320"],"corresponding_institution_ids":["https://openalex.org/I135310074"],"apc_list":null,"apc_paid":null,"fwci":5.4346,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.95816573,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"78","issue":"268","first_page":"2283","last_page":"2305"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9991000294685364,"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"}},"topics":[{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9991000294685364,"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"}},{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9976999759674072,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6255066990852356},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.396864116191864},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3961185812950134}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6255066990852356},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.396864116191864},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3961185812950134}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-09-02249-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02249-2","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02249-2/S0025-5718-09-02249-2.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:digitalcommons.butler.edu:facsch_papers-1098","is_oa":false,"landing_page_url":"https://digitalcommons.butler.edu/facsch_papers/98","pdf_url":null,"source":{"id":"https://openalex.org/S4306401403","display_name":"Digital Commons @ Butler University (Butler University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I110678235","host_organization_name":"Butler University","host_organization_lineage":["https://openalex.org/I110678235"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scholarship and Professional Work - LAS","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.160.1097","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.1097","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/mcom/2009-78-268/S0025-5718-09-02249-2/S0025-5718-09-02249-2.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-09-02249-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02249-2","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02249-2/S0025-5718-09-02249-2.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":[{"id":"https://openalex.org/G5998591216","display_name":"Aspects of Algorithmic Number Theory","funder_award_id":"0635355","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8143599214","display_name":"QnTM:    Physically-inspired Quantum Algorithms for NP-intermediate Problems","funder_award_id":"0523680","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"},{"id":"https://openalex.org/F4320306555","display_name":"Wisconsin Alumni Research Foundation","ror":"https://ror.org/00hwxbz16"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2045047334.pdf","grobid_xml":"https://content.openalex.org/works/W2045047334.grobid-xml"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W745330690","https://openalex.org/W1512503533","https://openalex.org/W1515416301","https://openalex.org/W1515528048","https://openalex.org/W1522524969","https://openalex.org/W1546982254","https://openalex.org/W1551619147","https://openalex.org/W1576065923","https://openalex.org/W1965610753","https://openalex.org/W1965792094","https://openalex.org/W1970072749","https://openalex.org/W1985203234","https://openalex.org/W1993449345","https://openalex.org/W1997459600","https://openalex.org/W1998458331","https://openalex.org/W1999877277","https://openalex.org/W2012407419","https://openalex.org/W2013169723","https://openalex.org/W2039343706","https://openalex.org/W2048513764","https://openalex.org/W2074590072","https://openalex.org/W2079428063","https://openalex.org/W2094438648","https://openalex.org/W2104281151","https://openalex.org/W2155673213","https://openalex.org/W2159341052","https://openalex.org/W2166717942","https://openalex.org/W2170413523","https://openalex.org/W2325057771","https://openalex.org/W2617116203","https://openalex.org/W2752853835","https://openalex.org/W2801179766","https://openalex.org/W2986262373","https://openalex.org/W3175367423","https://openalex.org/W4230514591","https://openalex.org/W4231463924","https://openalex.org/W4232018211","https://openalex.org/W4240867252","https://openalex.org/W4242849907","https://openalex.org/W4246279429","https://openalex.org/W4256082111","https://openalex.org/W4402262041"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W303980170","https://openalex.org/W2351491280","https://openalex.org/W2386767533","https://openalex.org/W2371447506","https://openalex.org/W3141679561","https://openalex.org/W4225307033"],"abstract_inverted_index":{"We":[0,160],"discuss":[1],"a":[2],"method":[3],"for":[4,119],"computing":[5,120],"<inline-formula":[6,54,84,124],"content-type=\"math/mathml\">":[7,55,85,125],"<mml:math":[8,56,86,126],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[9,57,87,127],"alttext=\"sigma-summation":[10],"Underscript":[11],"p":[12],"less-than-or-equal-to":[13],"x":[14,130],"Endscripts":[15],"1":[16,90],"slash":[17,61,91],"p\">":[18],"<mml:semantics>":[19,63,93,132],"<mml:mrow>":[20,133],"<mml:munder>":[21],"<mml:mo>":[22,28],"\u2211":[23],"</mml:mo>":[24,30],"<mml:mrow":[25,35,66,69,96,99],"class=\"MJX-TeXAtom-ORD\">":[26,36,67,70,97,100],"<mml:mi>p</mml:mi>":[27,39],"\u2264":[29],"<mml:mi>x</mml:mi>":[31,65,95,139],"</mml:mrow>":[32,38,40,72,74,102,104,142],"</mml:munder>":[33],"<mml:mn>1</mml:mn>":[34,98],"<mml:mo>/</mml:mo>":[37,71,101],"<mml:annotation":[41,76,106,143],"encoding=\"application/x-tex\">\\sum":[42],"_{p":[43],"\\le":[44],"x}":[45],"1/p</mml:annotation>":[46],"</mml:semantics>":[47,78,108,146],"</mml:math>":[48,79,109,147],"</inline-formula>":[49,80,110,148],",":[50,149],"using":[51],"time":[52],"about":[53,83],"alttext=\"x":[58,88],"Superscript":[59,89],"2":[60],"3\">":[62,92],"<mml:msup>":[64,94],"<mml:mn>2</mml:mn>":[68],"<mml:mn>3</mml:mn>":[73,103],"</mml:msup>":[75,105],"encoding=\"application/x-tex\">x^{2/3}</mml:annotation>":[77],"and":[81,153,158],"space":[82],"encoding=\"application/x-tex\">x^{1/3}</mml:annotation>":[107],".":[111],"It":[112],"is":[113],"based":[114],"on":[115],"the":[116,121,166,171],"Meissel-Lehmer":[117],"algorithm":[118,163],"prime-counting":[122],"function":[123],"alttext=\"pi":[128],"left-parenthesis":[129],"right-parenthesis\">":[131],"<mml:mi>":[134],"\u03c0":[135],"</mml:mi>":[136],"<mml:mo":[137,140],"stretchy=\"false\">(</mml:mo>":[138],"stretchy=\"false\">)</mml:mo>":[141],"encoding=\"application/x-tex\">\\pi":[144],"(x)</mml:annotation>":[145],"which":[150,170],"was":[151],"adapted":[152],"improved":[154],"by":[155],"Lagarias,":[156],"Miller,":[157],"Odlyzko.":[159],"used":[161],"this":[162],"to":[164],"determine":[165],"first":[167,175],"point":[168],"at":[169],"prime":[172],"harmonic":[173],"sum":[174],"crosses":[176],"4.":[177]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
