  Integrated time-slice correlation functions $G(t)$ with weights $K(t)$
appear, e.g., in the moments method to determine $\alpha_s$ from heavy quark
correlators, in the muon g-2 determination or in the determination of smoothed
spectral functions.
  For the (leading-order-)normalised moment $R_4$ of the pseudo-scalar
correlator we have non-perturbative results down to $a=10^{-2}$ fm and for
masses, $m$, of the order of the charm mass in the quenched approximation. A
significant bending of $R_4$ as a function of $a^2$ is observed at small
lattice spacings.
  Starting from the Symanzik expansion of the integrand we derive the
asymptotic convergence of the integral at small lattice spacing in the free
theory and prove that the short distance part of the integral leads to
$\log(a)$-enhanced discretisation errors when $G(t)K(t) \sim\, t $ for small
$t$. In the interacting theory an unknown, function $K(a\Lambda)$ appears.
  For the $R_4$-case, we modify the observable to improve the short distance
behavior and demonstrate that it results in a very smooth continuum limit. The
strong coupling and the $\Lambda$-parameter can then be extracted. In general,
and in particular for $g-2$, the short distance part of the integral should be
determined by perturbation theory. The (dominating) rest can then be obtained
by the controlled continuum limit of the lattice computation.
