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The finiteness condition of renormalization gives a restriction on the formof the gravitational action. By reconsidering the Hathrell's RG equations formassless QED in curved space, we determine the gravitational counterterms andthe conformal anomalies as well near four dimensions. As conjectured forconformal couplings in 1970s, we show that at all orders of the perturbationthey can be combined into two forms only: the square of the Weyl tensor in $D$dimensions and $E_D=G_4 +(D-4)\chi(D)H^2 -4\chi(D) \nabla^2 H$, where $G_4$ isthe usual Euler density, $H=R/(D-1)$ is the rescaled scalar curvature and$\chi(D)$ is a finite function of $D$ only. The number of the dimensionlessgravitational couplings is also reduced to two. $\chi(D)$ can be determinedorder by order in series of $D-4$, whose first several coefficients arecalculated. It has a universal value of $1/2$ at $D=4$. The familiar ambiguous$\nabla^2 R$ term is fixed. At the $D \to 4$ limit, the conformal anomaly $E_D$just yields the combination $E_4=G_4-2\nabla^2 R/3$, which induces Riegert'seffective action.

Determination of Gravitational Counterterms Near Four Dimensions from RG Equations