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, 8 TeV, and 7 TeV 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 → ?$^{+}$?$^{－}$ where ? = e, μ 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 → e$^{+}$ e$^{－}$ and the Z → μ$^{+}$ μ$^{－}$ 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 <
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