On the distribution of cyclic number fields of prime degree
type: Item
instance of: scholarly article
zbMATH Open document ID: 1280.11072
DOI: 10.1142/S1793042112500868
author: Seok Hyeong Lee
author: Gyujin Oh
published in: International Journal of Number Theory
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 6078182
zbMATH Keywords: cyclic number fields of prime degree
zbMATH Keywords: counting discriminants of number fields
zbMATH Keywords: generalized Riemann Hypothesis
MaRDI profile type: Publication
cites work: Exact computation of the discriminants of Abelian extensions
cites work: Counting discriminants of number fields
full work available at URL: s1793042112500868
OpenAlex ID: W2097389490
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title: On the distribution of cyclic number fields of prime degree
review text: Let \(N_{C_{p}}(X)\) denote the number of \(C_{p}\) Galois extensions of \(Q\) with absolute discriminant \(\leq X\). The work of \textit{D. J. Wright} [Proc. Lond. Math. Soc., III. Ser. 58, No. 1, 17--50 (1989; Zbl 0628.12006)] implies that when \(p\) is prime, we have NEWLINE\[NEWLINE N_{C_{p}}(X) = c(p)X^{1/(p-1)} + O(X^{1/p})NEWLINE\]NEWLINE for some positive \(c(p)\). NEWLINENEWLINEIn this paper, the authors, reduce the secondary error term to \(O(X^{1/2(p-1)})\). Moreover, under the Generalized Riemann Hypothesis, they obtain the following stronger result NEWLINE\[NEWLINE N_{C_{p}}(X) = c(p)X^{1/(p-1)} + X^{1/3(p-1)}R_{p}(\log\, X) + O(X^{\frac{1}{4(p-1)}+\epsilon})NEWLINE\]NEWLINE where \(R_{p}(X) \in Q[X]\) is polynomial of degree \(\lfloor p(p-2)/3\rfloor - 1\). This confirms an idea of \textit{H. Cohen, F. Diaz y Diaz} and \textit{M. Olivier} [J. Théor. Nombres Bordx. 18, No. 3, 573--593 (2006; Zbl 1193.11109)] in the case of \(C_{3}\) extensions.
reviewed by: Abdelmalek Azizi
description: scientific article; zbMATH DE number 6078182