Dirichlet Series and Asymptotics for Generalized Legendre Factorials

We introduce a Dirichlet-series framework for studying the asymptotic behavior of generalized factorial functions defined by Legendre-type valuation formulas. Let $K$ be a number field and let $S$ be a finite set of prime ideals. For a function $f$ on the prime ideals of $K\setminus S$, we define a factorial $n{!}_{K,f}$ by prescribing valuations $$v_{\mathfrak{p}}(n{!}_{K,f})=\sum _{k\ge 0}\lfloor \frac{n}{f(\mathfrak{p})\mathrm{N}({\mathfrak{p}}^k)}\rfloor \mathrm{.}$$ Using Perron’s formula and contour shifting, we obtain $$\log n{!}_{K,f}=a_{K,f,S}n\log n+C_{K,f,S}n+O\text{\hspace{0.17em}⁣}\left(n{e}^{-c\sqrt{\log n}}\right),$$ for some constants $a_{K,f,S},C_{K,f,S}$ up to a possible secondary term arising from an exceptional zero of ${\zeta }_K(s)$. The method applies naturally to rings of $S$-integers and provides an analytic explanation for the asymptotics of Legendre-type factorial constructions. As a result, we give asymptotics on a class of factorials with subsets in Dedekind domains finitely generated as $\mathbb{Z}$-algebras, partially answering a question of Bhargava on Stirling’s formula for his generalized factorials.