On the lengths of factorizations of elements in an algebraic number ring (Q1208166)

Let \(R\) be a Dedekind domain with finite ideal class-group \(G\) and let \(D(G)\) be the Davenport constant of \(G\). Assume moreover that every ideal-class contains at least one prime ideal. For any integer \(n\) denote by \(\Phi(n)\) the number of distinct integers \(m\) such that there exist irreducibles \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ n,\beta_ 1,\beta_ 2,\dots,\beta_ m\in R\) with \(\alpha_ 1\alpha_ 2\dots\alpha_ n=\beta_ 1\beta_ 2\dots\beta_ m\). The authors prove \[ \lim_{n\to\infty} {\Phi(n) \over n} =\;{{D(G)^ 2-4} \over {2D(G)}}. \]