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.