Farey lines defining Farey diagrams and application to some discrete structures

The aim of the paper is to study some of the analytical properties of Farey diagrams of order (m,n), which are associated to the (m,n)-cubes, that is the pieces of discrete planes, occurring in discrete mathematics. We give a closed formula for the number of Farey lines defining Farey diagrams. This number asymptotically behaves as mn(m+n)=ζ(3). Then we establish the relation with some discrete structures in the field of discrete geometry in particular.