  Using all the archival XMM-Newton X-ray (3-10 keV) observations of the
ultraluminous X-ray source (ULX) M82 X-1 we searched for a correlation between
its variable mHz quasi-periodic oscillation (QPO) frequency and its hardness
ratio (5-10 keV/3-5 keV), an indicator of the energy spectral power-law index.
When stellar-mass black holes (StMBHs) exhibit Type-C low-frequency QPOs (~
0.2-15 Hz) the centroid frequency of the QPO is known to correlate with the
energy spectral index. The detection of such a correlation would strengthen the
identification of M82 X-1's mHz QPOs as Type-C and enable a more reliable mass
estimate by scaling its QPO frequencies to those of Type-C QPOs in StMBHs of
known mass. We resolved the count rates and the hardness ratios of M82 X-1 and
a nearby bright ULX (source 5/X42.3+59) through surface brightness modeling. We
detected QPOs in the frequency range of 36-210 mHz during which M82 X-1's
hardness ratio varied from 0.42-0.47. Our primary results are: (1) we do not
detect any correlation between the mHz QPO frequency and the hardness ratio (a
substitute for the energy spectral power-law index) and (2) similar to some
accreting X-ray binaries, we find that M82 X-1's mHz QPO frequency increases
with its X-ray count rate (Pearson's correlation coefficient = +0.97). The
apparent lack of a correlation between the QPO centroid frequency and the
hardness ratio poses a challenge to the earlier claims that the mHz QPOs of M82
X-1 are the analogs of the Type-C low-frequency QPOs of StMBHs. On the other
hand, it is possible that the observed relation between the hardness ratio and
the QPO frequency represents the saturated portion of the correlation seen in
Type-C QPOs of StMBHs -- in which case M82 X-1's mHz QPOs can still be
analogous to Type-C QPOs.
