{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:50:45Z","timestamp":1776804645731,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"316","license":[{"start":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T00:00:00Z","timestamp":1558137600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["426694"],"award-info":[{"award-number":["426694"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We establish some new congruences satisfied by the Lind Mahler measure on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -groups, and use them to determine the Lind-Lehmer constant for many finite groups. First, we determine the minimal nontrivial measure of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -groups where one component has particularly high order. Second, we describe an algorithm that determines a small set of possible values for the minimal nontrivial measure of a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -group of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript p Baseline times double-struck upper Z Subscript p Sub Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msup>\n                                  <mml:mi>p<\/mml:mi>\n                                  <mml:mi>k<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_p\\times \\mathbb {Z}_{p^k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This algorithm is remarkably effective: applying it to more than 600000 groups the minimum was determined in all but six cases. Finally, we employ the results of our calculations to compute the Lind-Lehmer constant for nearly\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"8\">\n                        <mml:semantics>\n                          <mml:mn>8<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">8<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    million additional\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -groups.\n                  <\/p>","DOI":"10.1090\/mcom\/3350","type":"journal-article","created":{"date-parts":[[2018,4,4]],"date-time":"2018-04-04T09:51:27Z","timestamp":1522835487000},"page":"949-972","source":"Crossref","is-referenced-by-count":5,"title":["The Lind-Lehmer constant for certain \ud835\udc5d-groups"],"prefix":"10.1090","volume":"88","author":[{"given":"Dilum","family":"De Silva","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Mossinghoff","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vincent","family":"Pigno","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher","family":"Pinner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,5,18]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1307\/mmj\/1488510033","article-title":"(\ud835\udc5d-1)th roots of unity \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc5d\u207f, generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients","volume":"66","author":"Cochrane, Todd","year":"2017","journal-title":"Michigan Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-2285","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1515\/FORUM.2009.031","article-title":"Mahler measure under variations of the base group","volume":"21","author":"Dasbach, Oliver T.","year":"2009","journal-title":"Forum Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0933-7741","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1935","DOI":"10.1090\/S0002-9939-2014-11954-X","article-title":"The Lind Lehmer constant for \u2124_{\ud835\udd61}\u207f","volume":"142","author":"DeSilva, Dilum","year":"2014","journal-title":"Proc. Amer. Math. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"79","DOI":"10.4064\/aa142-1-7","article-title":"On the Lehmer constant of finite cyclic groups","volume":"142","author":"Kaiblinger, Norbert","year":"2010","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"461","DOI":"10.2307\/1968172","article-title":"Factorization of certain cyclotomic functions","volume":"34","author":"Lehmer, D. H.","year":"1933","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"5","key":"9","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1090\/S0002-9939-04-07753-6","article-title":"Lehmer\u2019s problem for compact abelian groups","volume":"133","author":"Lind, Douglas","year":"2005","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"10","unstructured":"M. Mossinghoff, V. Pigno, and C. Pinner, The Lind-Lehmer constant for \u2124\u2082^{\ud835\udd63}\u00d7\u2124\u2084^{\ud835\udd64}, preprint, arXiv:1805.05450[math.NT]."},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s11139-012-9443-1","article-title":"The Lind-Lehmer constant for cyclic groups of order less than 892,371,480","volume":"33","author":"Pigno, Vincent","year":"2014","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"key":"12","first-page":"Paper No. A46, 12","article-title":"The Lind-Lehmer constant for \u2124\u2098\u00d7\u2124\u207f_{\ud835\udd61}","volume":"16","author":"Pigno, Vincent","year":"2016","journal-title":"Integers"},{"key":"13","unstructured":"W. Vipismakul, The stabilizer of the group determinant and bounds for Lehmer\u2019s conjecture on finite abelian groups, Ph.D. Thesis, University of Texas at Austin, 2013."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03350-6\/mcom3350_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03350-6\/S0025-5718-2018-03350-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03350-6\/S0025-5718-2018-03350-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:59:37Z","timestamp":1776801577000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03350-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,18]]},"references-count":13,"journal-issue":{"issue":"316","published-print":{"date-parts":[[2019,3]]}},"alternative-id":["S0025-5718-2018-03350-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3350","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2018,5,18]]}}}