Dirichlet Series and Asymptotics for Generalized Legendre Factorials

Abstract: 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 $$ v{\mathfrak p}(n!{K,f})=\sum{k\geq 0}\left\lfloor \frac{n}{f(\mathfrak p)\mathrm{N}(\mathfrak p^k)}\right\rfloor. $$ Using Perron's formula and contour shifting, we obtain $$ \log n!{K,f} = a{K,f,S}n\log n + C_{K,f,S}n + O\, !\bigl(ne^{-c\sqrt{\log n}}\bigr), $$ for some constants $a_{K,f,S}, C_{K,f,S}$ up to a possible secondary term arising from an exceptional zero of $ζ_K(s)$. 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.