Class number 1 criteria for real quadratic fields of Richaud-Degert type
The real quadratic field \(\mathbb{Q} (\sqrt d)\) is called ``Richaud-Degart type'' if in the unique expression \(d = n^2 + r\) \((- n < r \leq n)\), \(r \mid 4n\) holds. In this paper, the authors intend to provide various class number one criteria for real quadratic fields of Richaud-Degert type by using two ways of computing special values of the zeta function \(\zeta(s)\) attached to the real quadratic field. In one of them, which is obtained by specializing Siegel's formula, the value \(\zeta (-1)\) is expressed in terms of the sum of divisors of integers. In another way, which is due to \textit{H. Lang} [cf. J. Reine Angew. Math. 233, 123-175 (1968; Zbl 0165.36504)], the value \(\zeta (-1, {\mathcal A})\) corresponding to an ideal class \({\mathcal A}\) is expressed in terms of the generalized Dedekind sums. By comparing these values they obtain various class number one criteria, which are expressed in terms of \(n\) and \(r\), for real quadratic fields of Richaud-Degert type.
real quadratic fields of Richaud-Degert type
scientific article; zbMATH DE number 874200