We consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence …