Dirichlet Series and Asymptotics for Generalized Legendre Factorials

1. Problem Statement

The paper addresses the asymptotic behavior of generalized factorial functions defined on rings of integers of number fields (and more generally, rings of $S$-integers).

• Context: The classical Legendre formula gives the $p$-adic valuation of $n!$ as $v_p(n!) = \sum_{k \ge 0} \lfloor n/p^k \rfloor$. This has been generalized to Dedekind domains where the denominators $p^k$ are replaced by arithmetic sequences attached to prime ideals.
• Specific Problem: The authors investigate the growth rate of a generalized factorial $n!_{K,f}$ defined by a valuation rule:
  $$v_p(n!_{K,f}) = \sum_{k \ge 0} \left\lfloor \frac{n}{f(p)N(p)^k} \right\rfloor$$
  where $K$ is a number field, $S$ is a finite set of prime ideals, $N(p)$ is the absolute norm of a prime ideal $p$, and $f$ is a function mapping primes to positive integers.
• Goal: To derive a precise asymptotic expansion for $\log(n!_{K,f})$ (or its summatory function) and provide a conceptual analytic explanation for these growth rates, partially answering a question posed by Bhargava regarding Stirling's formula for generalized factorials.

2. Methodology

The authors employ a Dirichlet series framework combined with complex analytic techniques, specifically Perron's formula and contour shifting.

A. Definition of the Increment Function

Instead of analyzing $n!_{K,f}$ directly, the authors study the logarithmic increment:
$$B_{K,f,S}(n) = \log \left( \frac{n!_{K,f}}{(n-1)!_{K,f}} \right)$$
They establish that $B_{K,f,S}(n)$ can be expressed as a sum over prime ideals:
$$B_{K,f,S}(n) = \sum_{p \notin S} \log N(p) \sum_{k \ge 0} \mathbb{1}_{f(p)N(p)^k \mid n}$$
where $\mathbb{1}_{m \mid n}$ is the indicator function that $m$ divides $n$.

B. Dirichlet Series Representation

The core innovation is constructing the Dirichlet series associated with $B_{K,f,S}(n)$:
$$D_{K,f,S}(s) = \sum_{n \ge 1} \frac{B_{K,f,S}(n)}{n^s}$$
Through absolute convergence arguments and rearranging sums, they derive the factorization:
$$D_{K,f,S}(s) = \zeta(s) \cdot H_{K,f,S}(s)$$
where $\zeta(s)$ is the Riemann zeta function and $H_{K,f,S}(s)$ is a series involving the prime ideals of $K$.

C. Connection to Dedekind Zeta Functions

The authors analyze $H_{K,f,S}(s)$ under the assumption that $f(p) \approx c N(p)$. They show that $H_{K,f,S}(s)$ is closely related to the logarithmic derivative of the Dedekind zeta function of $K$ (excluding $S$), denoted $-\frac{\zeta'_{K,S}}{\zeta_{K,S}}(s)$.
Specifically, they decompose $H_{K,f,S}(s)$ as:
$$H_{K,f,S}(s) = c^{-s} \left( -\frac{\zeta'_{K,S}}{\zeta_{K,S}}(s) \right) + J_{K,f,S}(s)$$
where $J_{K,f,S}(s)$ is a holomorphic function in a half-plane $\Re(s) > 1-\delta$.

D. Analytic Continuation and Pole Analysis

By combining the above, the Dirichlet series $D_{K,f,S}(s)$ is shown to possess a double pole at $s=1$.

• $\zeta(s)$ has a simple pole at $s=1$.
• $-\frac{\zeta'_{K,S}}{\zeta_{K,S}}(s)$ has a simple pole at $s=1$ (due to the pole of $\zeta_{K,S}$).
• The product creates a double pole, which dictates the leading asymptotic terms.

E. Asymptotic Derivation

Using Perron's formula, the summatory function $S_{K,f,S}(x) = \sum_{n \le x} B_{K,f,S}(n)$ is expressed as a contour integral. The authors shift the contour to the left, crossing the pole at $s=1$ and potentially an exceptional real zero $\beta_K$ of $\zeta_K(s)$. The residue at $s=1$ yields the main asymptotic terms, while the shifted contour provides the error term.

3. Key Contributions

1. Unified Analytic Framework: The paper provides a systematic Dirichlet-series approach to generalized factorials, moving beyond ad-hoc combinatorial or Chebyshev-function analyses used in previous works.
2. Pole Structure Identification: It rigorously proves that the Dirichlet series for the increment of these generalized factorials has a double pole at $s=1$. This explains why the asymptotic expansion contains a term of the form $x \log x$ (from the double pole) rather than just $x$.
3. Generalization to $S$-Integers: The results apply to rings of $S$-integers in any number field, covering Dedekind domains finitely generated as $\mathbb{Z}$-algebras.
4. Refined Error Terms: The method yields a sharp error term of the form $O(x e^{-c\sqrt{\log x}})$, which is superior to many previous estimates.

4. Main Results

Theorem 1.1 (Main Asymptotic Formula):
Let $K$ be a number field, $S$ a finite set of primes, and $f$ a function on primes such that $f(p) = c N(p) + O(N(p)^{1-\delta})$. The summatory function of the logarithmic increment satisfies:
$$S_{K,f,S}(x) = \frac{1}{c} x \log x + C_{K,f,S} x + O\left(x e^{-c_1 \sqrt{\log x}}\right)$$

• Main Term: $\frac{1}{c} x \log x$.
• Secondary Term: $C_{K,f,S} x$ (a constant depending on $K, f, S$).
• Exceptional Term: If the Dedekind zeta function $\zeta_K(s)$ has an exceptional real zero $\beta_K \in (0,1)$, an additional term $C^*_K x^{\beta_K}$ appears.
• Error Term: $O(x e^{-c_1 \sqrt{\log x}})$, derived from the zero-free region of $\zeta_K(s)$.

Corollary 7.2 (Application to Bhargava's Question):
The result applies to factorials defined on Dedekind domains finitely generated as $\mathbb{Z}$-algebras (specifically those with "Legendre subsets"). It confirms the existence of constants $a_{D,S}$ and $C_{D,S}$ such that:
$$\log n!_{D,S} = a_{D,S} n \log n + C_{D,S} n + O(n e^{-c_1 \sqrt{\log n}})$$
This provides a partial affirmative answer to Bhargava's inquiry about Stirling's formula for generalized factorials.

5. Significance

• Conceptual Clarity: The paper demystifies the growth of generalized factorials by linking them directly to the analytic properties of Dedekind zeta functions. The "double pole" phenomenon is identified as the fundamental reason for the $n \log n$ growth rate.
• Generality: The framework is robust enough to handle arbitrary number fields and variations in the valuation function $f$, provided $f$ grows linearly with the norm of the prime.
• Precision: By utilizing the zero-free region of the Dedekind zeta function, the authors achieve a very strong error bound, improving upon previous results that relied on less precise estimates.
• Bridging Fields: It connects multiplicative number theory (factorials), algebraic number theory (rings of integers, zeta functions), and analytic number theory (Dirichlet series, Perron's formula) in a cohesive manner.

In summary, Diaz and Normanyo successfully establish that the asymptotic behavior of Legendre-type generalized factorials is governed by the pole structure of the associated Dirichlet series, which in turn is determined by the Dedekind zeta function of the underlying number field.