The floor function strips away the fractional part of a number. It turns continuous quantities into discrete ones — and that is exactly what counting problems need.
A lattice point is a point with integer coordinates. Counting how many lie under a curve connects geometry, the floor function, and divisor sums in a single idea.
The area under 1/x from 1 to n is log n. The number of lattice points under n/x is approximately n log n. The gap between the two is the deepest open problem in analytic number theory.
If you sort all entries of an n×n multiplication table, the median is not near n²/2. It is much smaller. The reason is the hyperbola — and it makes the Dirichlet divisor problem concrete.
How does the sum of all divisor counts up to n grow? Dirichlet's answer — n log n with a precise error — is 175 years old. The sharp error bound is still open.