We discuss the calculation of the leading hadronic vacuum polarization in
lattice QCD. Exploiting the excellent quality of the compiled experimental data
for the e^+e^- --> hadrons cross-section, we predict the outcome of
large-volume lattice calculations at the physical pion mass, and design
computational strategies for the lattice to have an impact on important
phenomenological quantities such as the leading hadronic contribution to
(g-2)mu and the running of the electromagnetic coupling constant. First, the
R(s) ratio can be calculated directly on the lattice in the threshold region,
and we provide the formulae to do so with twisted boundary conditions. Second,
the current correlator projected onto zero spatial momentum, in a Euclidean
time interval where it can be calculated accurately, provides a potentially
critical test of the experimental R(s) ratio in the region that is most
relevant for (g-2)mu. This observation can also be turned around: the vector
correlator at intermediate distances can be used to determine the lattice
spacing in fm, and we make a concrete proposal in this direction. Finally, we
quantify the finite-size effects on the current correlator coming from
low-energy two-pion states and provide a general parametrization of the vacuum
polarization on the torus.Comment: 16 pages, 9 figure files; corrected a factor 2 in Eq. (7) over the
published versio