  The electromagnetic coupling constant, $\alpha$, is one of the fundamental
parameters of the Standard Model (SM). Its value at the Z boson mass,
$\alpha(M_Z)$, is of particular interest as it enters electroweak precision
tests. When running $\alpha$ from low energies up to the Z mass, five orders of
magnitude in precision are lost. This makes it one of the least well determined
parameters of the SM at that scale. The largest source of error comes from
non-perturbative hadronic effects in the low energy region. These
non-perturbative effects can be determined from ab-initio calculations in
lattice QCD. At higher energies, needed to match onto QCD perturbation theory,
discretization errors become large. In addition, the hadronic vacuum
polarization receives logarithmically-enhanced cutoff effects which render the
continuum extrapolation more difficult. To better control this extrapolation at
higher energies, we test a number of improvement procedures based on lattice
perturbation theory. To illustrate their effect, we present a preliminary
analysis of the light quark, connected contribution to the Adler function at
Euclidean $Q^2=5$ GeV$^2$. The lattice results are obtained using simulations
with $2+1+1$ flavors of staggered fermions at physical values of the quark
masses.
