  We show that for four dimensional gauge theories in the conformal window, the
Euler anomaly, known as the $a$-function, can be computed from a $2$-point
function of the trace of the energy momentum tensor making it more amenable to
lattice simulations. Concretely, we derive an expression for the $a$-function
as an integral over the renormalisation scale of quantities related to $2$- and
$3$-point functions of the trace of the energy momentum tensor.The crucial
ingredients are that the square of the field strength tensor is an exactly
marginal operator at the Gaussian fixed point and that the relevant $3$-point
correlation function is finite when resummed to all orders. This allows us to
define a scheme for which the $3$-point contribution vanishes, thereby
explicitly establishing the strong version of the $a$-theorem for this class of
theories.
