Partial Dedekind zeta values for ideal classes of the real quadratic field \(\mathbb{Q}(\sqrt{9m^2+2m})\) (Q6622522)

Let \(m\equiv 2\) (mod 3) be a positive integer such that \(D=9m^2+2m\) is squarefree: it follows that \(m\) is odd and \(D\equiv 3\) (mod 4). For these values of \(D\), the authors study the partial Dedekind zeta functions \(\zeta_k(s,C)\) for \(k=\mathbb{Q}(\sqrt{D})\) real quadratic field and \(C\) a class in the class group of \(k\). By employing the Generalized Dedekind sums as in [\textit{H. Lang}, J. Reine Angew. Math. 233, 123--175 (1968; Zbl 0165.36504)], [\textit{D. Byeon} and \textit{H. K. Kim}, J. Number Theory 62, No. 2, 257--272 (1997; Zbl 0871.11076)] and [\textit{N. K. Mahapatra} et al., Ramanujan J. 61, No. 3, 779--798 (2023; Zbl 1515.11109)], they are able to compute the special values \(\zeta_k(-1,C)\) when \(C\) is one of the following classes: the class of principal ideals, the class containing the unique prime factor of \(2\mathbb{Z}[\sqrt{d}]\), the class containing a prime factor of \(3\mathbb{Z}[\sqrt{d}]\) and, assuming \(p>3\) divides \(m\), the class including the unique prime factor of \(p\mathbb{Z}[\sqrt{d}]\).\N\NThis work follows completely the lines of the previously cited papers and of [\textit{H. Sankari} and \textit{A. Issa}, Int. J. Math. Math. Sci. 2020, 4 p. (2020; Zbl 1486.11131)], with no further developments in the employed techniques; the results on the class number \(h_k\) of the considered fields are not explicitly stated, but the results on the partial zeta function suggest a lower bound for \(h_k\) depending on the number of prime factors of \(D\).