What's the true regularized value of product of all natural numbers? Muñoz Garcia and Pérez-Marco - The product over all primes is $4\pi^2$ claims that the regularized value of product $\prod_{k=1}^\infty k$ is $\sqrt{2\pi}$ and of $\prod_{k=1}^\infty p_k$ over primes $p_k$ is $4\pi^2$.
This seems misleading to me because what they in fact calculate is the exponent of the regularized value of the logarithm of the product: $\exp(\operatorname{reg}\sum_{k=0}^\infty \ln a_k)$. I call this thing "hypermodulus", and it is more like determinant as opposed to regularized value which is more like trace (scalar part).
Thus the question arises, what are the true regularized values of these products? In my impression, the regularized value of $\prod_{k=1}^\infty k$ should be equal to the regularized value of the series $\sum_{k=1}^\infty (k-1)\Gamma(k)$.
 A: 
Zeta-regularization of arithmetic sequences by 
Jean-Paul Allouche (2020) provides a comprehensive discussion.

Is it possible to give a reasonable value to the infinite
product $1 \times 2 \times 3 \times \cdots \times n \times \cdots$? In
other words, can we define some sort of convergence of the finite
product $1 \times 2 \times 3 \times \cdots \times n$ when $n$ goes to
infinity? One way is to relate this product to the Riemann zeta
function and to its analytic continuation. This approach leads to: $1
\times 2 \times 3 \times \cdots \times n \times \cdots = \sqrt{2\pi}$.
More generally, the "zeta-regularisation" of an infinite product
consists of introducing a related Dirichlet series and its analytic
continuation at 0 (if it exists). We will survey some properties of
this generalized product and allude to applications.

A: The command of Mathematica 13.1
Product[j, {j, 2, Infinity}, Regularization -> "Dirichlet"]

produces $\sqrt{2 \pi }$. In fact, MMA finds the sum of the series of $\log k$, making use of analytic continuation of Dirichlet series.
