{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:09:28Z","timestamp":1786118968405,"version":"build-2736575974"},"reference-count":48,"publisher":"American Mathematical Society (AMS)","issue":"362","license":[{"start":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T00:00:00Z","timestamp":1787616000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n The Euler\u2013Kronecker constant of a number field\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function\n \n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n K<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\zeta _K(s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n at\n \n \n \n \n s<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n s=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We study the distribution of the Euler\u2013Kronecker constant\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n +<\/mml:mo>\n <\/mml:msubsup>\n \\gamma _q^+<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the maximal real subfield of\n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03be\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbb Q(\\xi _q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n \u03be\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n \\xi _q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -th root of unity, as\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ranges over the primes. Further, we consider the distribution of\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n +<\/mml:mo>\n <\/mml:msubsup>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\gamma _q^+-\\gamma _q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n \\gamma _q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the Euler\u2013Kronecker constant of\n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03be\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbb Q(\\xi _q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and show how it is connected with Kummer\u2019s conjecture, which predicts the asymptotic growth of the relative class number of\n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03be\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbb Q(\\xi _q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n +<\/mml:mo>\n <\/mml:msubsup>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\gamma _q^+-\\gamma _q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The methods employed are partly inspired by those used by Granville [Invent. Math. 100 (1990), pp. 321\u2013338] and Croot and Granville [J. London Math. Soc. (2) 66 (2002), pp. 579\u2013591] to investigate Kummer\u2019s conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.\n <\/p>","DOI":"10.1090\/mcom\/4134","type":"journal-article","created":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T18:58:55Z","timestamp":1756148335000},"page":"2991-3040","source":"Crossref","is-referenced-by-count":0,"title":["Relative class numbers and Euler\u2013Kronecker constants of maximal real cyclotomic subfields"],"prefix":"10.1090","volume":"95","author":[{"given":"Neelam","family":"Kandhil","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alessandro","family":"Languasco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pieter","family":"Moree","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sumaia","family":"Saad Eddin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alisa","family":"Sedunova","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,8,25]]},"reference":[{"key":"1","series-title":"Undergraduate Texts in Mathematics","volume-title":"Introduction to analytic number theory","author":"Apostol, Tom M.","year":"1976"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1134\/S0001434610010050","article-title":"The Euler-Kronecker constant","volume":"87","author":"Badzyan, A. 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