# Summing discrete coordinate lengths, a generalized n-dimensional case

A previous question pertains to a formula for the total number of points in a 3d discrete coordinate system that each total number of coordinate digit lengths $$l$$ can describe.

Is it possible to construct such a formula for a general $$n$$-dimensional case?

The coordinate digit length is defined as follows

$$l=\lfloor log(x_1) \rfloor + ... + \lfloor log(x_n) \rfloor+n$$.

where $$x_1,...,x_n\in$$ natural numbers.

In other words, I am looking for a formula that outputs the total number of positive integer points in a $$n$$-dimensional space that a certain total coordinate digit length can describe. For example, how many points can a total coordinate digit length of $$5$$ describe in a $$10$$-dimensional space?