I have not put much effort into this question but I have thought about it for a year or so. Is there such thing as a "logarithmic determinant"? The starting point for this is that the determinant of the Redheffer matrix gives the Mertens function in number theory. One can take a subset of the Redheffer matrix and then get each term of the Möbius function as a determinant.
Since the Dirichlet series for the Möbius function can be defined as a binomial series and the logarithm of the Riemann zeta function has a Dirichlet series that can be defined from a Taylor series, my question is:
Matrix inversion is to determinants as matrix logarithm is to what?