  The ratio $R(E)$ of the cross-sections for $e^+e^-\to$ hadrons and $e^+e^-\to
\mu^+\mu^-$ is a valuable energy-dependent probe of the hadronic sector of the
Standard Model. Moreover, the experimental measurements of $R(E)$ are the
inputs of the dispersive calculations of the leading hadronic vacuum
polarization contribution to the muon $g-2$ and these are in significant
tension with direct lattice calculations and with the muon $g-2$ experiment. In
this talk we discuss the results of our first-principles lattice study of
$R(E)$. By using a recently proposed method for extracting smeared spectral
densities from Euclidean lattice correlators, we have calculated $R(E)$
convoluted with Gaussian kernels of different widths $\sigma$ and central
energies up to $2.5$ GeV. Our theoretical results have been compared with the
KNT19 [1] compilation of experimental results smeared with the same Gaussian
kernels and a tension (about three standard deviations) has been observed for
$\sigma\sim 600$ MeV and central energies around the $\rho$-resonance peak.
