Let $A$ be a set of natural numbers.
Let $F(x,A)$ denote the number of natural numbers $n \le x$ divisible by no elements of $A$.
The Heilbronn-Rohrback Inequality says:
$$\Delta(A)=\lim_{x\to\infty}\frac{F(x,A)}{x} \ge \prod_{a\in A}(1-1/a) $$
Does this tell us anything about the value of $F(x,A)$? Can this be used to establish a lower bound for $F(x,A)$?
I've read the Wikipedia article on natural density and if this relationship exists, I'm not clear why it is not called out. On the other hand, if it doesn't have this relationship, I am unclear how the density inequality provides insights into the properties of natural numbers.
Let $L$ be the least common multiple of $A$. An integer $n$ is divisible by no elements in $A$ iff $n+L$ is divisible by no elements in $A$. This $L$-periodicity will give us some bounds: $$(x-L)\Delta(A)\leq F(x,A)\leq (x+L)\Delta(A)$$ (You'll want to double-check that; I haven't proved it, but it looks right.) Then applying the Heilbronn-Rohrback Inequality gives us this lower bound: $$F(x,A)\geq (x-L)\prod_{a\in A}(1-1/a)$$