I would like to prove (or disprove) the following assertions made regarding the Gamma function:
$$\lim _{z\rightarrow w}\Biggl(\frac{\mathcal A_{k} \left( \Gamma \left( z \right) \right) (3(1-\delta(k,1))+1) z^{k}(\frac{1}{z})^{z}\operatorname{e}^{z}}{\sqrt {\frac{2\pi}{z}}}\Biggr)=\mathcal P_{k}(w)=\sum _{j=0}^{k}\alpha_{{j,k}}{w}^{j} \quad\quad\,\,\,\,\,\,\,\,\,\,\,\,\,(0)$$ Where:
$\alpha_{{j,k}} \in \mathbb Z$
$\mathcal A_{k} \left( f\left( z \right) \right)$ is the asymptotic expansion of $f(z)$ computed to $k$ terms
$\delta \left( x,y \right) =\cases{1&$x=y$\cr 0&$x\neq y$\cr}$
In trying to deepen my understanding of the Gamma function and it's relationship with the prime numbers, I made a few conjectures based on the patterns I had observed aside from what I have already stated regarding (0).
As I have stated,polynomial $\mathcal P_{k}(z)$, that appears as a factor of the asymptotic expansion of the gamma function, (where k is the number of terms the expansion was computed)has integer coefficients, and the sum of these coefficients always has a maximum integer factor that is prime, i.e in it's unique prime factorization product, the largest term always has a multiplicity of 1:
$$\max\Biggl( \mathcal F\Bigl(\sum _{j=0}^{k}\alpha_{{j,k}}\Bigr)\Biggr) \in \mathbb P\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,(1)$$ where $\mathcal F(N)$ a set whose elements are the terms of the unique factorization product of $N$. Furthermore, the set of coefficients for $\mathcal P_{k+1}(z)$ and the set of coefficients of $\mathcal P_{k}(z)$ are disjoint, or in general we have:
For $ \mathbb A_{k}={\{\alpha_{{j,k}}}\}_{j=0..k}$ :
$$\mathbb A_{k_i} \cap\mathbb A_{k_j} ={\{\,}\}\,\,\, \forall k_i,k_j \in \mathbb N\,\,\,\operatorname{s.t}\,\, k_i \neq k_j\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$$
Regarding the first assertion, for the first 1000 natural numbers, roughly just over 84 percent of them have largest factors that are prime, and the remaining 16 percent are composite. Of the first $10^6$ natural numbers, 96 percent have prime maximum factors. I find it therefore worthy of observation that these values defined in (1) are always prime.
But the percentage of numbers less than or equal to $n$ having a prime maximum factor asymptotically will approach 100 percent, this presenting us with a bit of a paradox, seeing we know already there exist numbers with composite maximum factors, making it impossible for this value to ever truly be reached!
As an additional side note,$k=1$ has as a $\alpha_{k}$ set equal to the empty product, and the corresponding polynomial is $\mathcal P_1(z)=z$,which implies the empty product is equal to unity, ie this result gives credence to propositions that non numerical objects in fact are natural numbers computationally speaking:
$${\{\{\,\,}\}\}=1\in \mathbb N\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(3)$$