The elementary Selberg sieve and bounded prime gaps

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This post is a continuation of the previous post on sieve theory, which is an ongoing part of the Polymath8 project to improve the various parameters in Zhang’s proof that bounded gaps between primes occur infinitely often. Given that the comments on that page are getting quite lengthy, this is also a good opportunity to “roll over” that thread.

We will continue the notation from the previous post, including the concept of an admissible tuple, the use of an asymptotic parameter $latex {x}&fg=000000$ going to infinity, and a quantity $latex {w}&fg=000000$ depending on $latex {x}&fg=000000$ that goes to infinity sufficiently slowly with $latex {x}&fg=000000$, and $latex {W := prod_{p<w} p}&fg=000000$ (the $latex {W}&fg=000000$-trick).

The objective of this portion of the Polymath8 project is to make as efficient as possible the connection between two types of results, which we call $latex {DHL[k_0,2]}&fg=000000$ and $latex {MPZ[varpi,delta]}&fg=000000$. Let us first state $latex…

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