Using linear algebra to study number theory?

Question

I've posted a paper on arXiv that outlines a linear algebra approach to number theory.

Specifically, I have the following questions:

1. Is it possible to draw connections between the factorization matrix (Def. 4 and 6) and the Redheffer matrix or the Mertens function? Does this allow to come up with an alternative formulation of the Riemann hypothesis (Eq. 55 and 56)?

2. Is it possible to evaluate the difference $\pi(x) - \mathrm{Li}(x)$ using the matrix algebra expression for $\pi(x)$ (Eq. 44)? How can the inverse of the factorization matrix (Eq. 39) be further developped to help in this respect?

Answers to either one of the questions are appreciated!