We conclude the rigorous analysis of a previous paper [V. Moretti, Commun. Math. Phys. 201, 327 (1999)] concerning the relation between the (Euclidean) point-splitting approach and the local ζ-function procedure to renormalize physical quantities at one-loop in (Euclidean) Quantum Field Theory in curved space–time. The case of the stress tensor is now considered in general D-dimensional closed manifolds for positive scalar operators −Δ+V(x). Results obtained formally in previous works [in the case D=4 and V(x)=ξR(x)+m2] are rigorously proven and generalized. It is also proven that, in static Euclidean manifolds, the method is compatible with Lorentzian-time analytic continuations. It is proven that the result of the ζ-function procedure is the same obtained from an improved version of the point-splitting method which uses a particular choice of the term w0(x,y) in the Hadamard expansion of the Green’s function, given in terms of heat-kernel coefficients. This version of the point-splitting procedure works for any value of the field mass m. If D is even, the result is affected by an arbitrary one-parameter class of (conserved in absence of external source) symmetric tensors, dependent on the geometry locally, and it gives rise to the general correct trace expression containing the renormalized field fluctuations as well as the conformal anomaly term. Furthermore, it is proven that, in the case D=4 and V(x)=ξR(x)+m2, the given procedure reduces to the Euclidean version of Wald’s improved point-splitting procedure provided the arbitrary mass scale present in the ζ-function is chosen opportunely. It is finally argued that the found point-splitting method should work generally, also dropping the hypothesis of a closed manifold, and not depending on the ζ-function procedure. This fact is indeed checked in the Euclidean section of Minkowski space–time for A=−Δ+m2 where the method gives rise to the correct Minkowski stress tensor for m2⩾0 automatically.

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