We show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity. We determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large.

As a consequence, we provide an alternative proof of a conjecture posed by Bar and Schopka concerning the convergence of the determinant of the Dirac operator on , determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the proof by Moller.