{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:50:57Z","timestamp":1773244257062,"version":"3.50.1"},"reference-count":19,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2016,10,1]],"date-time":"2016-10-01T00:00:00Z","timestamp":1475280000000},"content-version":"unspecified","delay-in-days":274,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2016]]},"abstract":"The Li coefficients $\\unicode[STIX]{x1D706}_{F}(n)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of a zeta or $L$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-function $F$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\\unicode[STIX]{x1D70F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport\u2013Heilbronn zeta function. The behavior of the $\\unicode[STIX]{x1D70F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\\text{Re}(z)>\\unicode[STIX]{x1D70F}\/2.$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and $\\unicode[STIX]{x1D70F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> aspects.<\/jats:p>","DOI":"10.1112\/s1461157016000115","type":"journal-article","created":{"date-parts":[[2016,10,4]],"date-time":"2016-10-04T07:12:30Z","timestamp":1475565150000},"page":"259-280","source":"Crossref","is-referenced-by-count":6,"title":["On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients"],"prefix":"10.1112","volume":"19","author":[{"given":"Alina","family":"Bucur","sequence":"first","affiliation":[]},{"given":"Anne-Maria","family":"Ernvall-Hyt\u00f6nen","sequence":"additional","affiliation":[]},{"given":"Almasa","family":"Od\u017eak","sequence":"additional","affiliation":[]},{"given":"Lejla","family":"Smajlovi\u0107","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2016,10,1]]},"reference":[{"key":"S1461157016000115_r14","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157010000215"},{"key":"S1461157016000115_r11","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2311"},{"key":"S1461157016000115_r13","article-title":"On the \ud835\udf0f-Li coefficients for automorphic L-functions","author":"Mazhouda","journal-title":"Rocky Mountain J. Math.,"},{"key":"S1461157016000115_r15","doi-asserted-by":"publisher","DOI":"10.1090\/S1061-0022-2013-01242-8"},{"key":"S1461157016000115_r2","first-page":"15","article-title":"Around Davenport\u2013Heilbronn function","volume":"66","author":"Bombieri","year":"2011","journal-title":"Uspekhi Mat. Nauk"},{"key":"S1461157016000115_r3","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1999.2392"},{"key":"S1461157016000115_r4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-17987-2_7"},{"key":"S1461157016000115_r12","first-page":"103","article-title":"Li\u2019s criterion for the Riemann hypothesis \u2014 numerical approach","volume":"24","author":"Maslanka","year":"2004","journal-title":"Opuscula Math."},{"key":"S1461157016000115_r9","first-page":"133","volume-title":"Analytic number theory, C.I.M.E. Summer School, Cetraro, Italy, 2002","author":"Kaczorowski","year":"2006"},{"key":"#cr-split#-S1461157016000115_r6.1","unstructured":"1. A. D. Droll, 'Variations of Li's criterion for an extension of the Selberg class', PhD Thesis, Queen's University Ontario, Canada, 2012"},{"key":"#cr-split#-S1461157016000115_r6.2","unstructured":"2. available at http:\/\/qspace.library.queensu.ca\/jspui\/bitstream\/1974\/7352\/1\/Droll_Andrew_D_201207_PhD.pdf."},{"key":"S1461157016000115_r17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2009.10.012"},{"key":"S1461157016000115_r18","volume-title":"The theory of the Riemann zeta-function","author":"Titchmarsh","year":"1951"},{"key":"S1461157016000115_r5","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-11.4.307"},{"key":"S1461157016000115_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2015.03.019"},{"key":"S1461157016000115_r16","first-page":"367","volume-title":"Proceedings of Amalfi Conference on Analytic Number Theory","author":"Selberg","year":"1992"},{"key":"S1461157016000115_r10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392574"},{"key":"S1461157016000115_r1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-01999-0"},{"key":"S1461157016000115_r8","doi-asserted-by":"publisher","DOI":"10.1145\/2576802.2576828"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157016000115","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T20:07:29Z","timestamp":1555704449000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157016000115\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016]]}},"alternative-id":["S1461157016000115"],"URL":"https:\/\/doi.org\/10.1112\/s1461157016000115","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]}}}