Classical Cepheid period-Wesenheit-metallicity relation in the Gaia
  bands
  Classical Cepheids (DCEPs) represent a fundamental tool to calibrate the
extragalactic distance scale. However, they are also powerful stellar
population tracers in the context of Galactic studies. The forthcoming Data
Release 3 (DR3) of the Gaia mission will allow us to study, with unprecedented
detail, the structure, the dynamics, and the chemical properties of the
Galactic disc, and in particular of the spiral arms, where most Galactic DCEPs
reside. In this paper, we aim to quantify the metallicity dependence of the
Galactic DCEPs' period-Wesenheit ($PWZ$) relation in the Gaia bands. We adopted
a sample of 499 DCEPs with metal abundances from high-resolution spectroscopy,
in conjunction with Gaia Early Data Release 3 parallaxes and photometry to
calibrate a $PWZ$ relation in the Gaia bands. We find a significant metallicity
term, of the order of $-$0.5 mag/dex, which is larger than the values measured
in the near-infrared (NIR) bands by different authors. Our best $PWZ$ relation
is $W=(-5.988\pm0.018)-(3.176\pm0.044)(\log P-1.0)-(0.520\pm0.090){\rm
[Fe/H]}$. We validated our $PWZ$ relations by using the distance to the Large
Magellanic Cloud as a benchmark, finding very good agreement with the geometric
distance provided by eclipsing binaries. As an additional test, we evaluated
the metallicity gradient of the young Galactic disc, finding $-0.0527 \pm
0.0022$ dex/kpc, which is in very good agreement with previous results.
