  Ratios of top-quark pair to $Z$-boson cross sections measured from
proton--proton collisions at the LHC centre-of-mass energies of $\sqrt
s=13$TeV, 8TeV, and 7TeV are presented by the ATLAS Collaboration. Single
ratios, at a given $\sqrt s$ for the two processes and at different $\sqrt s$
for each process, as well as double ratios of the two processes at different
$\sqrt s$, are evaluated. The ratios are constructed using previously published
ATLAS measurements of the $t\overline{t}$ and $Z$-boson production cross
sections, corrected to a common phase space where required, and a new analysis
of $Z \rightarrow \ell^+ \ell^-$ where $\ell=e,\mu$ at $\sqrt s=13$TeV
performed with data collected in 2015 with an integrated luminosity of
$3.2$fb$^{-1}$. Correlations of systematic uncertainties are taken into account
when evaluating the uncertainties in the ratios. The correlation model is also
used to evaluate the combined cross section of the $Z\rightarrow e^+e^-$ and
the $Z\rightarrow \mu^+ \mu^-$ channels for each $\sqrt s$ value. The results
are compared to calculations performed at next-to-next-to-leading-order
accuracy using recent sets of parton distribution functions. The data
demonstrate significant power to constrain the gluon distribution function for
the Bjorken-$x$ values near 0.1 and the light-quark sea for $x<0.02$.
