Quantitative relation between the regularized functional and the second Dirichlet-series derivative

Establish, if such a relation exists, a quantitative identity linking the evaluation functional \(F=\sum_{k=2}^{\infty}\Lambda_k/k^2\), which defines the monoid regularization of \(P_{\omega}=\prod_{n\ge1}n^{(-1)^{\omega(n)}}\), to the second derivative \(D_{\omega}''(1)=8C_2\ln 2\) of its associated Dirichlet series.

Background

The paper associates the divergent arithmetic product $P_{\omega}=\prod_{n\ge1}n^{(-1)^{\omega(n)}}$ with the Dirichlet series $D_{\omega}(s)=\sum_{n\ge1}(-1)^{\omega(n)}n^{-s}$. Its generalized-monoid construction produces the absolutely convergent evaluation functional $F=\sum_{k=2}^{\infty}\Lambda_k/k^2$, whose exponential is taken as the regularized value of the product.

Independently, the paper proves the analytic identity $D_{\omega}''(1)=8C_2\ln 2$, where $C_2$ is the Hardy–Littlewood twin-prime constant. The authors explicitly state that no quantitative identity currently connects these two quantities and leave the existence of such a relation unresolved.