  Population synthesis models predict that high-mass X-ray binary (HMXB)
populations produced in low metallicity environments should be more X-ray
luminous, a trend supported by studies of nearby galaxies. This trend may be
responsible for the observed increase of the X-ray luminosity
($L_{\mathrm{X}}$) per star formation rate (SFR) with redshift due to the
decrease of metallicity ($Z$) at fixed stellar mass as a function of redshift.
To test this hypothesis, we use a sample of 79 $z\sim2$ star-forming galaxies
with oxygen abundance measurements from the MOSDEF survey, which obtained
rest-frame optical spectra for $\sim1500$ galaxies in the CANDELS fields at
$1.37<z<3.80$. Using Chandra data from the AEGIS-X Deep, Deep Field North, and
Deep Field South surveys, we stack the X-ray data at the galaxy locations in
bins of redshift and $Z$ because the galaxies are too faint to be individually
detected. In agreement with previous studies, the average $L_{\mathrm{X}}$/SFR
of our $z\sim2$ galaxy sample is enhanced by $\approx0.4-0.8$ dex relative to
local HMXB $L_{\mathrm{X}}$-SFR scaling relations. Splitting our sample by $Z$,
we find that $L_{\mathrm{X}}$/SFR and $Z$ are anti-correlated with 97%
confidence. This observed $Z$ dependence for HMXB-dominated galaxies is
consistent both with the local $L_{\mathrm{X}}$-SFR-$Z$ relation and a subset
of population synthesis models. Although the statistical significance of the
observed trends is weak due to the low X-ray statistics, these results
constitute the first direct evidence connecting the redshift evolution of
$L_{\mathrm{X}}$/SFR and the $Z$ dependence of HMXBs.
