Far-reaching statistical consequences of the zero-point energy for the harmonic oscillator

In a recent thermodynamic analysis of the harmonic oscillator and using an interpolation procedure, Boyer has shown that the existence of a zero-point energy leads to the Planck spectrum. Here we avoid the interpolation by adding a statistical argument to arrive at Planck's law as an inescapable result of the presence of the zero-point energy. No explicit quantum argument is introduced along the derivations. We disclose the connection of our results with the original analysis of Planck and Einstein, which led to the notion of the quantized radiation field. We then inquire into the discrete or continuous behaviour of the energy and pinpoint the discontinuities. Finally, to open the door to the description of the zero-point fluctuations, we briefly discuss the statistical (in contrast to the purely thermodynamic) description of the oscillator, which accounts for both thermal and temperature-independent contributions to the energy dispersion.