Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan (Ramanujan J.) (20230101), 61, no.~3, 779-798. ISSN: 1382-4090 (print).eISSN: 1572-9303.
Subject
11 Number theory -- 11R Algebraic number theory: global fields 11R11 Quadratic extensions
In stark contrast with the imaginary quadratic fields, the class numbers of real quadratic fields are still highly mysterious. As the problem of infinitude of real quadratic fields with class number one is a fundamentally deep question, many special classes of real quadratic fields have been studied by several authors. One such class of real quadratic fields is the R-D type fields, namely the fields of the form $\Bbb{Q}(\sqrt{n^2+r})$ with $n, r\in\Bbb{Z}$, $-n < r \leq n$ and $r\mid 4n$. Class number one, two and three problems for this class of fields are well studied. In the paper under review, the authors consider the fields of the form $k\coloneq \Bbb{Q}(\sqrt{d})$, where $d=9m^2+4m$ is square-free and $2\nmid m$. They prove that for ${m\nequiv 2\pmod 3}$, the class number of $k$ is one if and only if $m=1, \pm 3$. They also prove that for $m\equiv 1 \pmod 3$ with $m\notin\{ -5, 1\}$, the class number of $k$ is at least 3. \par The main tool used is a computation and comparison of the Siegel formula as described by D.~B. Zagier [Enseign. Math. (2) {\bf 22} (1976), no.~1-2, 55--95; MR0406957] for the value of the complete Dedekind zeta function at $-1$ and the formula due to H. Lang [J. Reine Angew. Math. {\bf 233} (1968), 123--175; MR0238804] for the analogous value at $-1$ of the partial Dedekind zeta function corresponding to a certain fractional ideal. At the end of the paper, some computational evidence is given to demonstrate the results.