{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T07:29:03Z","timestamp":1776756543162,"version":"3.51.2"},"reference-count":38,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:p>\n            Keiper [1] and Li [2] published independent investigations of the connection between the Riemann hypothesis and the properties of sums over powers of zeros of the Riemann zeta function. Here we consider a reframing of the criterion, to work with higher-order derivatives\n            <jats:italic>\n              \u03be\n              <jats:sub>r<\/jats:sub>\n            <\/jats:italic>\n            of the symmetrized function\n            <jats:italic>\u03be<\/jats:italic>\n            (\n            <jats:italic>s<\/jats:italic>\n            ) at\n            <jats:italic>s<\/jats:italic>\n            = 1\/2, with all\n            <jats:italic>\n              \u03be\n              <jats:sub>r<\/jats:sub>\n            <\/jats:italic>\n            known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the\n            <jats:italic>\n              \u03be\n              <jats:sub>r<\/jats:sub>\n              .\n            <\/jats:italic>\n            This is studied using known asymptotic expansions for the\n            <jats:italic>\n              \u03be\n              <jats:sub>r<\/jats:sub>\n            <\/jats:italic>\n            , which give the location of the summand as a relationship between two parameters. This relationship needs to be inverted, which we show can be done exactly using a generalized Lambert function. The result enables an accurate asymptotic expression for the value of the infinite sum.\n          <\/jats:p>","DOI":"10.1145\/3637529.3637530","type":"journal-article","created":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T17:06:57Z","timestamp":1702487217000},"page":"85-110","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions"],"prefix":"10.1145","volume":"57","author":[{"given":"Ross","family":"McPhedran","sequence":"first","affiliation":[{"name":"School of Physics, University of Sydney, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tony C.","family":"Scott","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physikalische Chemie, RWTH-Aachen University, Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aude","family":"Maignan","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes, CNRS, Grenoble INP, LJK, Grenoble, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2023,12,13]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"765","article-title":"Power series expansions of Riemann's xi function","volume":"58","author":"Keiper J. 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General relativity and quantum mechanics: towards a generalization of the Lambert W function. Applicable Algebra in Engineering, Communication and Computing, 17(1):41--47, 2006.","journal-title":"Communication and Computing"},{"key":"e_1_2_1_17_1","doi-asserted-by":"crossref","unstructured":"T. C. Scott G. J. Fee and J. Grotendorst. Asymptotic series of generalized Lambert W function. SIGSAM 47(3):75--83 September 2013.","DOI":"10.1145\/2576802.2576804"},{"issue":"2","key":"e_1_2_1_18_1","first-page":"42","article-title":"Numerics of the generalized Lambert W function","volume":"48","author":"Scott T. C.","year":"2014","unstructured":"T. C. Scott, G. J. Fee, J. Grotendorst, and Wanzhou Zhang. Numerics of the generalized Lambert W function. 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