Dirichlet Series and Asymptotics for Generalized Legendre Factorials
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arXiv
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| Format: | Preprint |
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2026
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| author | Diaz, Brian Normanyo, Pascal |
| author_facet | Diaz, Brian Normanyo, Pascal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13720 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dirichlet Series and Asymptotics for Generalized Legendre Factorials Diaz, Brian Normanyo, Pascal Number Theory 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. |
| title | Dirichlet Series and Asymptotics for Generalized Legendre Factorials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.13720 |