Special values of zeta functions of the simplest cubic fields and their applications (Q1130266)

The so-called simplest cubic fields are the fields \(K\) defined by the polynomials \(x^3 + mx^2 -(m+3)x + 1\), where \(m \in \mathbb{Z}\) and \(m^2 +3m+9\) is square-free. Let \(\zeta(s,A)\) denote the partial zeta function of an absolute ideal class \(A\) of \(K\). Applying a result by \textit{U. Halbritter} and \textit{M. Pohst} [J. Number Theory 36, No. 3, 266-288 (1990; Zbl 0727.11050)], the author computes \(\zeta(2,A)\) for a special class \(A\). Two previously known results for the class number \(h_K\) are obtained as easy corollaries: (1) If \(h_K=1\), then \(2m+3\) is prime; (2) if \(2m+3=a^n\) for positive integers \(a\) and \(n\), then \(n\) divides \(h_K\).