Some variations on the Dedekind conjecture (Q2764526)

This paper deals with holomorphy of quotients of Artin \(L\)-functions and Dedekind zeta-functions. It gives a survey of results. It is not clear, however, whether the authors are aware that some of their stated results have been shown before. Lemma 2.1 is due to \textit{P. X. Gallagher} and proved in full in \textit{B. Huppert}'s book ``Endliche Grup\-pen'' (1967; Zbl 0217.02201)]. Lemma 2.2 is implicitly contained in [\textit{R. W. van der Waall}, Indagationes Math. 37, 83--86 (1975; Zbl 0298.12003)] worked out for \(\varphi=1_H\) there, and \textit{N. S. Hekster} (1975) mentioned the same proof holds verbatim for any 1-dimensional character \(\varphi\in L\text{Irr}(H)\) [in the paper under review \(\varphi\) has to be 1-dimensional, otherwise it is false in general]. Lemma 2.3, and more precise its way of proving, is clear and any specialist in the field knows for 30 years almost, that the semi-direct product situation is the crucial point. Lemma 2.4 is to be found as Theorem 4 in Indagationes Math., New Ser. 4, 99--109 (1993; Zbl 0779.11056): ``On a problem of R. Brauer for quotients of Dedekind Zeta-Functions'' by \textit{R. W. van der Waall} and \textit{K. Sato}. The proof printed there equals the proof given in the paper under review. The reviewer attributes the knowledge of Theorem 4 of that 1993 paper to R. Foote (1990, unpublished).NEWLINENEWLINE In the paper under review we find an extension (not a generalization) of the Uchida-van der Waall theorem as follows:NEWLINENEWLINE Theorem: Let \(K/F\) be a solvable extension of number fields, i.e. \(K/F\) is a Galois extension of number fields and \(G=\text{Gal}(K/F)\) is solvable. Let \(\chi\in\text{Char} (G)\). Then for every \(H\leq G\) and 1-dimensional character \(\psi\) of \(H\), if \(m(\chi,\psi)= \langle\chi,\text{Ind}^G_H\psi \rangle_G\), then NEWLINE\[NEWLINE\frac{L(s,\text{Ind}^G_H (\psi),K/F)} {L(s,\chi, K/F)^{m(\chi,\psi)}}NEWLINE\]NEWLINE is regular at \(s=s_0\neq 1\).NEWLINENEWLINE Summarising: the paper lacks a proper way of quoting, but it contains as a matter of fact a lot of nice results.