A method is given to obtain the Green's function for the Poisson equation in any arbitrary integer dimension under periodic boundary conditions. We obtain recursion relations which relate the solution in d-dimensional space to that in (d-1)-dimensional space. Near the origin, the Green's function is shown to split in two parts, one is the essential Coulomb singularity and the other part is regular. We are thus able to give representations of the Coulomb sum in higher dimensions without taking recourse to any integral representations. The expressions converge exponentially fast in all part of the simulation cell. Works of several authors are shown to be special cases of this more general method.