  The present calculations in perturbative QCD reach the order $\alpha_s^4$ for
several correlators calculated to five loops, and the huge computational
difficulties make unlikely the full six-loop calculation in the near future.
This situation has practical consequences, in particular the treatment of the
higher orders of the perturbation series for the current-current correlator of
light quarks is one of the main sources of errors in the extraction of the
strong coupling from hadronic $\tau$ decays. Several approximate estimates of
the next coefficients of the corresponding Adler function have been proposed,
using various arguments. In the present paper we exploit the analytic structure
of the Adler function in the Borel plane, which allows the definition of an
improved perturbative expansion in powers of a conformal variable which maps
the cut Borel plane onto the unit disk. The new expansions converge in a larger
domain of the Borel plane and, when reexpanded in powers of the strong
coupling, yield definite values for the higher perturbative coefficients. We
apply the method to the Adler function in the $\bar{\rm MS}$ scheme and to a
suitable weighted integral of this function in the complex $s$ plane, chosen
such as to avoid model-dependent assumptions on analyticity. Our results
$c_{5,1}=287 \pm 40$, $c_{6,1}=2948 \pm 208$ and $c_{7,1}=(1.89 \pm 0.75)\times
10^4$, for the six, seven and eigth-loop coefficients, respectively, agree with
a recent determination from Pad\'e approximants applied to the perturbative
expansion of the hadronic $\tau$ decay rate.
