The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends naturally usefully to discrete lattices such as the set of all integer lattice points in d-dimensional space. This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include extended connections between zeta-dimension and classical fractal dimensions, a gale characterization of zeta-dimension, and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers.