Connection between Brun's Sieve and the Sieve of Eratosthenes

Question

I have read in a couple of places that the above mentioned sieves are connected in some way. In particular, Brun's Sieve builds upon that of Eratosthenes. I do not see why this is the case and hope someone could explain the connection

Answer 1

Brun introduced several new ideas about sieves (and several sieves too..) that have pushed the collection of techniques with similar flavor known as sieve theory very far, and used them to partially answer some very hard questions.
The pure sieve is a good example to (probably) illustrate the connection you're looking for.
Here's a very general answer:

The pure sieve starts in the same place as the sieve of Eratosthenes.
We have some set $A$ of integers all $\le x$ and a subset $A_p\subset A$ which we want to sieve for, i.e. what's removed. If we want to sieve the numbers in several subsets $A_{p_1},..,A_{p_k}\subset A$ we use a certain sum with which we can determine the size of $A$ after the elements from those subsets were removed.

The sieve of Eratosthenes in "full glory" says, under certain technical assumptions, that the aforementioned sum is asymptotic to some (nice) functions which are the main term, and of course, there's an error term.

So, the actual ways Brun's pure sieve improves on that of Eratosthenes, technically, are:

1. It relaxes the assumptions in the sieve of Eratosthenes a little, and
2. It provides an asymptotic approximation (to the same sum we mentioned) with a much better error term.