In Appendix A, Polchinski does the Euclidean path integral for the Harmonic oscillator. After he Pauli-Villars regularizes the determinant of the kinetic term, he obtains the following expression (A.1.62):
$$ \langle q_f, U | q_i, 0 \rangle \to \left( \frac{\omega}{2 \sinh \omega U} \right)^{1/2} \exp \left[ - S_{cl}(q_i,q_f) + \frac{1}{2}\left( \Omega U - \ln \Omega \right) - S_{ct} \right] \tag{A.1.62}. $$
where $\Omega$ is a frequency scale.
In what follows he says:
"To get a finite answer as $\Omega \to \infty$, we need first to include a term $\frac{1}{2} \Omega$ in the Lagrangian $L_{ct}$, canceling the linear divergence in (A.1.62). That is, there is a linearly divergent bare coupling for the operator 1. It may seem strange that we need to renormalize in a quantum mechanics problem, but power counting is completely uniform with quantum field theory. The logarithmic divergence is a wavefunction renormalization.
What does the italic sentence mean? The operator $1$ should change by some $Z. 1$? Why the logarithm is a wave function renormalization? And is he power counting what?