{"id":"https://openalex.org/W3206330529","doi":"https://doi.org/10.1017/fmp.2026.10040","title":"A variational principle for Gaussian lattice sums","display_name":"A variational principle for Gaussian lattice sums","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W3206330529","doi":"https://doi.org/10.1017/fmp.2026.10040","mag":"3206330529"},"language":"en","primary_location":{"id":"doi:10.1017/fmp.2026.10040","doi":"https://doi.org/10.1017/fmp.2026.10040","is_oa":true,"landing_page_url":"https://doi.org/10.1017/fmp.2026.10040","pdf_url":null,"source":{"id":"https://openalex.org/S2911503828","display_name":"Forum of Mathematics Pi","issn_l":"2050-5086","issn":["2050-5086"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-3","norway-2","oarl","poland-70","russia-white-list-1"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Forum of Mathematics, Pi","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.1017/fmp.2026.10040","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060076655","display_name":"Laurent B\u00e9termin","orcid":"https://orcid.org/0000-0003-4070-3344","observed_orcids":["https://orcid.org/0000-0003-4070-3344"]},"institutions":[{"id":"https://openalex.org/I100532134","display_name":"Lyon 1 Universit\u00e9","ror":"https://ror.org/029brtt94","country_code":"FR","type":"education","lineage":["https://openalex.org/I100532134","https://openalex.org/I203339264"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Laurent B\u00e9termin","raw_affiliation_strings":["Universit\u00e9 Claude Bernard Lyon 1"],"raw_orcid":"https://orcid.org/0000-0003-4070-3344","affiliations":[{"raw_affiliation_string":"Universit\u00e9 Claude Bernard Lyon 1","institution_ids":["https://openalex.org/I100532134"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025959741","display_name":"Markus Faulhuber","orcid":"https://orcid.org/0000-0002-7576-5724","observed_orcids":["https://orcid.org/0000-0002-7576-5724"]},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Markus Faulhuber","raw_affiliation_strings":["University of Vienna"],"raw_orcid":"https://orcid.org/0000-0002-7576-5724","affiliations":[{"raw_affiliation_string":"University of Vienna","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054778975","display_name":"Stefan Steinerberger","orcid":"https://orcid.org/0000-0002-7745-4217","observed_orcids":["https://orcid.org/0000-0002-7745-4217"]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stefan Steinerberger","raw_affiliation_strings":["University of Washington"],"raw_orcid":"https://orcid.org/0000-0002-7745-4217","affiliations":[{"raw_affiliation_string":"University of Washington","institution_ids":["https://openalex.org/I201448701"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5025959741"],"corresponding_institution_ids":["https://openalex.org/I129774422"],"apc_list":{"value":2650,"currency":"USD","value_usd":2650},"apc_paid":{"value":2650,"currency":"USD","value_usd":2650},"fwci":12.361,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.942622,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":88,"max":97},"biblio":{"volume":"14","issue":null,"first_page":"307","last_page":"341"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.3391000032424927,"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":0.3391000032424927,"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.31470000743865967,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.2321999967098236,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/jacobi-theta-function","display_name":"Jacobi theta function","score":0.8510000109672546},{"id":"https://openalex.org/keywords/hexagonal-lattice","display_name":"hexagonal lattice","score":0.9200000166893005},{"id":"https://openalex.org/keywords/lattice-density","display_name":"lattice density","score":0.5989999771118164},{"id":"https://openalex.org/keywords/variational-principle","display_name":"variational principle","score":0.9229999780654907},{"id":"https://openalex.org/keywords/ionic-crystals","display_name":"ionic crystals","score":0.8050000071525574},{"id":"https://openalex.org/keywords/minimum-energy","display_name":"minimum energy","score":0.6470000147819519},{"id":"https://openalex.org/keywords/completely-monotone-functions","display_name":"completely monotone functions","score":0.9620000123977661},{"id":"https://openalex.org/keywords/landau-constants","display_name":"Landau constants","score":0.9929999709129333}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7308744192123413},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6924649477005005},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.6650683879852295},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6185932159423828},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5311388373374939},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5109609961509705},{"id":"https://openalex.org/C90626534","wikidata":"https://www.wikidata.org/wiki/Q4523381","display_name":"Hexagonal lattice","level":3,"score":0.4977555572986603},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.41897666454315186},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3904361128807068},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3701549172401428},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3695223331451416},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.21307402849197388},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.20585784316062927},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1986791491508484},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C155355069","wikidata":"https://www.wikidata.org/wiki/Q575224","display_name":"Antiferromagnetism","level":2,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1017/fmp.2026.10040","doi":"https://doi.org/10.1017/fmp.2026.10040","is_oa":true,"landing_page_url":"https://doi.org/10.1017/fmp.2026.10040","pdf_url":null,"source":{"id":"https://openalex.org/S2911503828","display_name":"Forum of Mathematics Pi","issn_l":"2050-5086","issn":["2050-5086"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-3","norway-2","oarl","poland-70","russia-white-list-1"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Forum of Mathematics, Pi","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2110.06008","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2110.06008","pdf_url":"https://arxiv.org/pdf/2110.06008","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-04454244v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/hal-04454244v1","pdf_url":"https://hal.science/hal-04454244v1/file/2110.06008.pdf","source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"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":"2024","raw_type":"info:eu-repo/semantics/preprint"},{"id":"mag:3206330529","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/pdf/2110.06008.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"pmh:oai:doaj.org/article:801ab9e2777348a9888b5bef6fd6afb1","doi":null,"is_oa":false,"landing_page_url":"https://doaj.org/article/801ab9e2777348a9888b5bef6fd6afb1","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,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Forum of Mathematics, Pi, Vol 14 (2026)","raw_type":"article"},{"id":"doi:10.48550/arxiv.2110.06008","doi":"https://doi.org/10.48550/arxiv.2110.06008","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2110.06008","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.1017/fmp.2026.10040","doi":"https://doi.org/10.1017/fmp.2026.10040","is_oa":true,"landing_page_url":"https://doi.org/10.1017/fmp.2026.10040","pdf_url":null,"source":{"id":"https://openalex.org/S2911503828","display_name":"Forum of Mathematics Pi","issn_l":"2050-5086","issn":["2050-5086"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-3","norway-2","oarl","poland-70","russia-white-list-1"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Forum of Mathematics, Pi","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1526607814","display_name":null,"funder_award_id":"2123224","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1633768630","display_name":null,"funder_award_id":"FR 4083/3-1/I 4354","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2374857920","display_name":null,"funder_award_id":"2123224","funder_id":"https://openalex.org/F4320306151","funder_display_name":"Alfred P. Sloan Foundation"},{"id":"https://openalex.org/G2578060530","display_name":"Geometric Properties of Second Order Elliptic Partial Differential Equations","funder_award_id":"2123224","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G411161539","display_name":"Inside the Frame Set","funder_award_id":"P 33217","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G519738200","display_name":null,"funder_award_id":"2123224","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G5442947621","display_name":"Variational Modeling of Molecular Geometries","funder_award_id":"I4354","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G668528873","display_name":null,"funder_award_id":"FR 4083/3-1/I 4354","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G6785396967","display_name":null,"funder_award_id":"FR 4083/3-1/I 4354","funder_id":"https://openalex.org/F4320306151","funder_display_name":"Alfred P. Sloan Foundation"},{"id":"https://openalex.org/G7126559616","display_name":null,"funder_award_id":"FR 4083/3-1/I","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G7207901211","display_name":null,"funder_award_id":"FR 4083/3-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G9000172126","display_name":null,"funder_award_id":"10.55776/P33217","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3206330529.pdf","grobid_xml":"https://content.openalex.org/works/W3206330529.grobid-xml"},"referenced_works_count":65,"referenced_works":["https://openalex.org/W97402757","https://openalex.org/W176966770","https://openalex.org/W652032100","https://openalex.org/W1487878339","https://openalex.org/W1556956362","https://openalex.org/W1584654650","https://openalex.org/W1593679773","https://openalex.org/W1605417594","https://openalex.org/W1644604237","https://openalex.org/W1671104699","https://openalex.org/W1971903277","https://openalex.org/W1985323904","https://openalex.org/W2001676859","https://openalex.org/W2024860340","https://openalex.org/W2028323766","https://openalex.org/W2032372625","https://openalex.org/W2032793159","https://openalex.org/W2033500205","https://openalex.org/W2034041690","https://openalex.org/W2039343706","https://openalex.org/W2040881478","https://openalex.org/W2041693988","https://openalex.org/W2056007801","https://openalex.org/W2061156351","https://openalex.org/W2067811790","https://openalex.org/W2071932383","https://openalex.org/W2090324061","https://openalex.org/W2091725015","https://openalex.org/W2100955645","https://openalex.org/W2109453208","https://openalex.org/W2109548554","https://openalex.org/W2119337797","https://openalex.org/W2125811087","https://openalex.org/W2147389341","https://openalex.org/W2160860771","https://openalex.org/W2289880074","https://openalex.org/W2327901725","https://openalex.org/W2414887269","https://openalex.org/W2520410766","https://openalex.org/W2571168512","https://openalex.org/W2587040628","https://openalex.org/W2588409462","https://openalex.org/W2607441980","https://openalex.org/W2745295315","https://openalex.org/W2762578211","https://openalex.org/W2797344977","https://openalex.org/W2798198027","https://openalex.org/W2798453318","https://openalex.org/W2807214083","https://openalex.org/W2906924634","https://openalex.org/W2908461350","https://openalex.org/W2920886339","https://openalex.org/W2944510906","https://openalex.org/W2945046525","https://openalex.org/W2954126175","https://openalex.org/W2969407857","https://openalex.org/W2973818247","https://openalex.org/W3021492718","https://openalex.org/W3046248970","https://openalex.org/W3046459323","https://openalex.org/W3100495456","https://openalex.org/W3103274847","https://openalex.org/W3156496371","https://openalex.org/W3216915741","https://openalex.org/W4366974727"],"related_works":["https://openalex.org/W3046248970","https://openalex.org/W3216915741","https://openalex.org/W4400201665","https://openalex.org/W4405494223","https://openalex.org/W4226035408","https://openalex.org/W2396540398","https://openalex.org/W2920886339","https://openalex.org/W4289560580","https://openalex.org/W2014685665","https://openalex.org/W4404988557"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"consider":[2],"a":[3],"two-dimensional":[4],"analog":[5],"of":[6,28],"Jacobi":[7],"theta":[8],"functions":[9],"and":[10],"prove":[11],"that,":[12],"among":[13],"all":[14],"lattices":[15],"\u039b":[16],"\u2282":[17],"R":[18,32],"2":[19],"$\\Lambda":[20],"\\subset":[21],"\\mathbb":[22],"{R}^2$":[23],"normal":[24],"upper":[25,31],"Lamda":[26],"subset":[27],"double":[29],"struck":[30],"squared":[33],"with":[34],"fixed":[35],"density,":[36],"the":[37,43,52,62,66],"minimal":[38],"value":[39],"is":[40],"maximized":[41],"by":[42],"hexagonal":[44,63],"lattice.":[45],"This":[46],"result":[47],"can":[48],"be":[49],"seen":[50],"as":[51],"inhomogeneous":[53],"counterpart":[54],"to":[55],"Montgomery\u2019s":[56],"1988":[57],"result,":[58],"which":[59],"proved":[60],"that":[61],"lattice":[64],"minimizes":[65],"maximum.":[67]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":1}],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2021-10-25T00:00:00"}
