On Dirichlet series of a certain commutative matrix ring (Q1382838)

This highly technical paper deals with subjects related to work of \textit{C. J. Bushnell} and \textit{I. Reiner} [Math. Z. 173, 135-161 (1980; Zbl 0438.12004) and J. Reine Angew. Math. 327, 156-183 (1981; Zbl 0455.12010)], and to work of \textit{C. G. Latimer} and \textit{C. C. MacDuffee} [Ann. Math. (2) 34, 313-316 (1933; Zbl 0006.29002)]. Zeta-functions and integral representation theory play a role too. This very well written paper is a toolkit for future research. The main theorems of the paper are the following; here \(\mathbb{C}\) denotes the complex number field. Theorem I. Let \(C= C({\mathfrak a})\) be the ideal class of the ring \(R=\mathbb{Z} [\zeta]\) represented by an integral ideal \({\mathfrak a}\). Define a Dirichlet series \(\zeta_C (s)\) for \(C\) by \[ \zeta_C (s)= \sum_{{\mathfrak b} \in C ({\mathfrak a}) \atop {\mathfrak b} \subset \mathbb{R}} {1\over (N{\mathfrak b})^s}, \quad s\in \mathbb{C}. \] Then \(\zeta_C (s)\) is holomorphic in the half plane \(\text{Re} (s)>1\). Furthermore, we have \[ \lim_{\sigma \to 1+0} (\sigma-1)^g \zeta_C (\sigma)= 2^{r +c} \pi^c {(\mathbb{E}_O: \mathbb{E}_{\mathfrak a}) \bigl| R (\mathbb{E}_O) \bigr| \over N \check {\mathfrak a} N{\mathfrak a} | H_O |\sqrt {| \mathbb{D} |}}. \] Here \(R (\mathbb{E}_O)\) is the regulator of \(\mathbb{E}_O\), \(H_O\) is the group of all elements in \(\mathbb{E}_O\) with finite order, \(r\) (resp. \(2c)\) is the number of all real (resp. complex) roots of the characteristic polynomial \(f(X)\) of \(\zeta\), \(g\) is the number of irreducible factors of \(f(X)\) over \(\mathbb{Z}\) and \(\mathbb{D} =Nf'(\zeta)\) is the discriminant of \(R\). Theorem II. We define a Dirichlet series \(\zeta_R (s)\) by \[ \zeta_R (s)= \sum_{\mathfrak b} {a({\mathfrak b}) \over (N{\mathfrak b})^s} \] where the summation runs over all nonsingular ideals \({\mathfrak b}\) of \(R\) and \(a({\mathfrak b})= N\check {\mathfrak b} N{\mathfrak b}/(\mathbb{E}_O:\mathbb{E}_{\mathfrak b})\). Then we have \[ \lim_{\sigma\to 1+0} (\sigma-1)^g \zeta_R (\sigma)= |\mathbb{G} |^{2r+c} \pi^c {\bigl | R(\mathbb{E}_O) \bigr| \over| H_O |\sqrt {| \mathbb{D} |}}. \] For unexplained notation, please consult the paper.