On the irreducibility in the ring of analytic Dirichlet series (Q558345)

With any sequence \( (a_n)_{n\geq 1} \) of complex coefficients, one can associate a formal Dirichlet series \( \sum_{n\geq 1}a_n n^{-s} \). Such a series is said to be analytic if it converges for some complex number \( s \). The set \( {\mathcal D}[[s]] \) (resp. \( {\mathcal D}\{s\} \)) of formal (resp. analytic) Dirichlet series is a ring for the natural operations on series. The present article is a continuation of the study of \( {\mathcal D}[[s]] \) and \( {\mathcal D}\{s\} \) from the viewpoint of analytic geometry and division techniques, which was initiated by the authors in previous papers [C. R., Math. Acad. Sci., Paris, 336, No. 3, 213--218 (2003; Zbl 1068.11056), Ann. Inst. Fourier 53, No. 7, 2039--2060 (2003; Zbl 1077.32002)]. They show that an element of \( {\mathcal D}\{s\} \) is irreducible in \( {\mathcal D}\{s\} \) if and only if it is irreducible in \( {\mathcal D}[[s]] \). In the same spirit, it is shown that \( {\mathcal D}\{s\} \) is integrally closed in \( {\mathcal D}[[s]] \). The last part of the paper is devoted to a tricky example of finitely generated ideal in \( {\mathcal D}\{s\}\) for which Buchberger's algorithm does not terminate. This ideal has no Gröbner basis.