  The partition function of four dimensional Euclidean, non-supersymmetric
SU(2) Yang--Mills theory is calculated in the perturbative and weak coupling
regime i.e. in a small open ball about the flat connection (what we call the
vicinity of the vacuum) and when the gauge coupling constant acquires a small
but finite value.
  The computation is based on various known inequalities, valid only in four
dimensions, providing two-sided estimates for the exponentiated Yang--Mills
action in terms of the $L^2$-norm of the derivative of the gauge potential
only; these estimates then give rise to Gaussian-like infinite dimensional
integrals involving the Laplacian hence can be formally computed via
zeta-function and heat kernel techniques. It then turns out that these
integrals give a sharp value for the partition function in the aforementioned
perturbative and weak coupling regime of the theory.
  In the resulting expression for the partition function the original classical
coupling constant is shifted to a smaller one which can be interpreted as the
manifestation, in this approach, of the existence a non-trivial trivial
$\beta$-function and asymptotic freedom in non-Abelian gauge theories.
