Dirichlet Series and Asymptotics for Generalized Legendre Factorials

TL;DR

This paper develops a Dirichlet-series approach to analyze the asymptotic behavior of generalized factorial functions defined via Legendre-type valuations over number fields, providing new insights and partial answers to questions about Stirling's formula generalizations.

Contribution

It introduces a Dirichlet-series framework for generalized factorials over number fields and derives precise asymptotics, extending classical factorial analysis to algebraic number theory contexts.

Findings

Derived asymptotic formulas for generalized factorials in number fields.

Connected factorial asymptotics to zeros of Dedekind zeta functions.

Partially answered Bhargava's question on Stirling's formula for generalized 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 $\zeta_K(s)$. 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.