Define a Dirichlet series $\zeta_X^D(s)$ by the relation $$ \left(-\log \zeta_X^D(s) \right)' \; \zeta(s) = \sum_{n\geq 1} \frac{\log\left( n!_X^D/(n-1)!_X^D\right)}{n^s}, $$ where $n!_X^D = |D/v_n(X,D)|$ is the Bhargava factorial for a Dedekind domain $D$ and a subset $X \subseteq D$, and $v_n(X,D)$ is the $n$-th factorial ideal as defined by Bhargava. If $D$ is known, we write for short-hand $n!_X$.
In deriving a Stirling's formula for $n!_X^D$, many special cases are known, specifically when $X = D$: these cases observe that a Stirling's formula for $n!_X^D$ is deeply connected with information about the poles and zeroes of $\zeta_X^D$ for special cases.
Conjecture (Stirling's formula for Dedekind domains finitely generated as $\mathbb{Z}$-algebras) If $D$ is a Dedekind domain finitely generated as a $\mathbb{Z}$-algebra, then $$n!_D^D \sim \sqrt{2\pi n}^{ \mathrm{rk} \mathrm{Cl}(D)-\mathrm{rk} \mathcal O_{\mathrm{Spec}(D)}^\times({\mathrm{Spec}(D)})}n^{\mathrm{Irr}(D)n}e^{-(\mathrm{Irr}(D)+\gamma_D-\gamma \mathrm{Irr}(D))n-S_D(n)+(B_D(n)+C_D)/2},$$ where $\mathrm{Irr}(D)$ is the number of irreducible components of $\mathrm{Spec}(D)$ with maximal dimension; $\mathrm{Cl}(D)$ is the ideal class group of $D$; $\mathcal O_{\mathrm{Spec}(D)}^\times({\mathrm{Spec}(D)})$ is the group of invertible regular functions over $\mathrm{Spec}(D)$; $\gamma_D$ and $C_D$ are constants; the sum $$\displaystyle S_D(n) = \sum_{\zeta_{\mathrm{Spec}(D)}(\rho) = 0,} \zeta(\rho)\frac{n^\rho}{\rho}, \, \text{ for } \mathrm{Re}(\rho) \in (0,1);$$ $B_D(n)$ is the $n$-th Dirichlet series coefficient of $-\zeta_{\mathrm{Spec}(D)}'(s)\zeta(s)/\zeta_{\mathrm{Spec}(D)}(s)$; and $\zeta_\mathfrak{X}$ is the arithmetic zeta function of a scheme $\mathfrak{X}$ of finite type over $\mathbb{Z}$.
Not much is known when $X \neq D$; thus, we are interested in determining more non-trivial examples of $X \subsetneq D$ for which $n!_X^D$ admits a Stirling's formula. For a start, let us consider $X = \mathbb{P}$, the set of primes, and $D = \mathbb{Z}$; then, using OEIS with sequence A202318, we have \begin{align*} \left(-\log \zeta_\mathbb{P}(s) \right)' \; \zeta(s) & =\sum_{n > 1} \log \mathfrak{B}_{2n-1}\left( \frac{1}{(2n)^s} - \frac{1}{(2n+1)^s}\right) - \frac{\log \mathrm{lcm}\{\mathfrak{B}_2, \mathfrak{D}_2\}}{2} \\ & - \sum_{n > 1} \log \mathrm{lcm}\{\mathfrak{B}_{2n}, \mathfrak{D}_{2n}\}\left( \frac{1}{(2(n+1))^s} - \frac{1}{(2n+1)^s}\right), \end{align*}
where
$$\mathfrak{B}_{n} = \prod_{p \, \text{prime}, \; s_p(n) \geq p} p, \hspace{1cm} \mathfrak{D}_{n} = \prod_{p-1 \, | \, n} p, $$
and $s_p(n)$ is the sum of the base $p$ digits of $n$. From [1], we have $\zeta_\mathbb{P}$ converges for $\mathrm{Re}(s) > 1$; then the asymptotic formulas for $\mathfrak{B}_n, \mathfrak{D}_n$ in [1] suggest we may analytically continue $\zeta_\mathbb{P}$ to $\mathrm{Re}(s) > 1/2$. Then, if $\zeta_\mathbb{P}(s) = \zeta(s)L_\mathbb{P}(s)$ for some Dirichlet series $L_\mathbb{P}(s)$, then, from [1], we find $L'_\mathbb{P}(1)/L_\mathbb{P}(1) = C$, where $$C = \sum_{p \in \mathbb{P}} \frac{\log p}{(p-1)^2}.$$
If $\zeta_\mathbb{P}(s)$ can be meromorphically continued to $\mathbb{C}$, we should expect the simple pole $s=1$ to come from $\zeta(s)$ and that $L_\mathbb{P}(s)$ may be entire.
However, I am having a difficult time finding resources giving stronger approximations on $\mathfrak{B}_n$ and $\mathfrak{D}_n$ in an effort to analytically continue $\zeta_\mathbb{P}$.
Question: Do we know of any stronger asymptotic formulas for $\mathfrak{B}_n$ and $\mathfrak{D}_n$? If not, what other methods are available to analytically continue a Dirichlet series, specifically $\zeta_\mathbb{P}$?
References
[1] Bordelles, O., et. al. Denominators of Bernoulli polynomials. https://arxiv.org/abs/1706.09804
