  We study the multiplicative Hilbert matrix, i.e. the infinite matrix with
entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently
introduced within the context of the theory of Dirichlet series, and it was
shown that the multiplicative Hilbert matrix has no eigenvalues and that its
continuous spectrum coincides with $[0,\pi]$. Here we prove that the
multiplicative Hilbert matrix has no singular continuous spectrum and that its
absolutely continuous spectrum has multiplicity one. Our argument relies on the
tools of spectral perturbation theory and scattering theory. Finding an
explicit diagonalisation of the multiplicative Hilbert matrix remains an
interesting open problem.
