Units in abelian group rings and meromorphic functions (Q1812654)

Let \(\mathbb{Z}\Gamma\) denote the integral group ring of the finite, abelian group \(\Gamma\), and \(\|\;\|\) denote any Euclidean norm on \(\mathbb{C}\Gamma\). The author continues his study of \[ \iota_{\|\;\|}(s)= \sum_{\substack{ x\in\mathbb{Z}\Gamma^\times\\ \| x\|>1}} (\log\| x\|)^{-s}, \qquad s\in \mathbb{C} \] [\textit{G. Everest}, J. Reine Angew. Math. 375/376, 24-41 (1987; Zbl 0601.12015)], where he showed that, for \(r_\Gamma\) the torsion free rank of \(\mathbb{Z}\Gamma^\times\), (i) \(\iota_{\|\;\|}(s)\) has half-plane of convergence \(\text{Re}(s)> r_\Gamma\); and (ii) \(\iota_{\|\;\|}(s)\) has analytic continuation to \(\text{Re}(s)> r_\Gamma-1\), where it is analytic apart from a simple pole at \(s=r_\Gamma\). The residue is independent of the choice of norm \(\|\;\|\). Let \(\widehat{\Gamma}\) denote the group of characters of \(\Gamma\), \(\widehat{\Gamma}= \Hom (\Gamma,\mathbb{C}^\times)\). These can be used to provide a useful example of an Euclidean norm. For each \(\chi\in \widehat{\Gamma}\), \(x= \sum_{\gamma\in \Gamma} x_\gamma \gamma\in\mathbb{C}\Gamma\), let \(\iota_\chi(x)= \sum_\gamma \chi(\gamma)x_\gamma\). Then \[ \| x\|=| x|= \max_{x\in\widehat{\Gamma}} \{|\iota_\chi (x)|\} \] gives an Euclidean norm. The author proves, in this paper, that the series \(\iota_{|\;|}(s)\) has analytic continuation to \(\text{Re}(s)> r_\Gamma-2\), and that the only singularities in this half-plane are simple poles at \(s=r_\Gamma\) and \(s=r_\Gamma-1\). Refinements of \(\iota_{\|\;\|}(s)\) are also studied. The author states that an application to Galois module theory, similar to that in [the author, Manuscr. Math. 57, 451-467 (1987; Zbl 0611.12008)], is obtainable.