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Dirichlet Series and Asymptotics for Generalized Legendre Factorials

Brian Diaz, Pascal Normanyo

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.

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 be a number field and let be a finite set of prime ideals. For a function on the prime ideals of , we define a factorial by prescribing valuations Using Perron's formula and contour shifting, we obtain for some constants up to a possible secondary term arising from an exceptional zero of . The method applies naturally to rings of -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 -algebras, partially answering a question of Bhargava on Stirling's formula for his generalized factorials.
Paper Structure (13 sections, 11 theorems, 95 equations)

This paper contains 13 sections, 11 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Rings of $S$-integers
  4. Dedekind zeta functions
  5. Analytic properties
  6. Perron's formula
  7. Notation
  8. Generalized Legendre Factorials
  9. The Associated Dirichlet Series
  10. Comparison with the Logarithmic Derivative of $\zeta_K(s)$
  11. Laurent Expansion of the Dirichlet Series
  12. Perron's Formula and the Proof of the Main Theorem
  13. Examples and Remarks

Key Result

Theorem 1.1

Let $K$ be a number field and $S$ a finite set of prime ideals of $K$. Suppose $f$ is a function on the prime ideals of $K\setminus S$ satisfying for some constants $c>0$ and $\delta>0$. Define the generalized factorial by and let Then the summatory function satisfies for some constants $C_{K,f,S}\in\mathbb R$ and $c_1>0$, up to a possible additional term $C_K^\ast x^{\beta_K}$ if the Dedekin

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Lemma 5.1
  • ...and 17 more