Lattice sums arising from the Poisson equation

In recent times, attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). Herein we study a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms ⏀_n(r₁,...,r_n) = 1/π²[formula could not be replicated]. By virtue of striking connections with Jacobi ϑ-function values, we are able to develop new closed forms for certain values of the coordinates r_k, and extend such analysis to similar lattice sums. A primary result is that for rational x, y, the natural potential ⏀²(x, y) is 1/π log A where A is an algebraic number. Various extensions and explicit evaluations are given. Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 20,000-digit arithmetic.