Dirichlet oscillator fluctuation determinant

The quadratic fluctuation eigenvalues are $(n\pi /\mathcal{T}{)}^2+{\omega }^2$. Dividing by the free-particle determinant gives ${\prod }_{n\ge 1}[1+(\omega \mathcal{T}/n\pi {)}^2{]}^{-1/2}=\sqrt{\omega \mathcal{T}/\sinh (\omega \mathcal{T})}$. The short-time free kernel fixes the absolute normalization.