Given $N$, is there a formula for $card( \{(m,n)\, s.t.\, m\cdot n \leq N \} )$?

Question

The formula is also equivalent to :

$$ \sum_{m=1}^N \left \lfloor \frac{N}{m} \right \rfloor $$

An interpretation would be to count the discrete rectangles with total area inferior to N. But aside from that, I don't really see how to express this...

Of course, it should not be hard to compute, but I was curious to see if there is a formula for this.

On the same topic, is the following equality true ? Basically the same formula with bounding both $m$ and $n$ (and not just their product). All variables are integers.

$$ \text{card}\left( \Big\{(m,k)\, s.t.\, m\cdot k \leq N, m\in[1,M] , k\in[1,K] \Big\}\right) = \sum_{m=1}^{\min(M,N)}\min\left(K,\left \lfloor \frac{N}{m} \right \rfloor\right)$$

Answer 1

This is called Dirichlet's divisor problem and is a famous problem in number theory. From Wikipedia:

The divisor summatory function is defined as $$D(x)=\sum_{n\le x} d(n) = \sum_{j,k \atop jk\le x} 1$$ where $$d(n)=\sigma_0(n) = \sum_{j,k \atop jk=n} 1$$ is the divisor function. The divisor function counts the number of ways that the integer n can be written as a product of two integers.

The formula with quasi-polynomials is not correct. In fact, nobody knows how to derive an explicit formula. Asymptotics have been studied, though.