I have been trying to understand what is exactly a sieve and why sieves are useful. I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they are useful.
Can someone explain what is a sieve in general terms and what properties of sieves make them so useful in attacking number theoretic problems? Are there general principles that one can use as a guideline to see if sieves are likely to be useful in attacking a problem?
I know this question might be a bit elementary for number theorists, but I wasn't able to find good and concise information at the level of a general mathematician. (The Wikipedia article links to Ben Green's notes about sieve theory but the link is not working anymore. I Googled a bit however I couldn't find a concise high-level but rigorous exposition, notes like this and this dive quickly into examples and applications without giving any general definition and discussion of sieves.)