Jurkat–Richert theorem


The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.
It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.

Statement of the theorem

This formulation is from Diamond & Halberstam.
Other formulations are in Jurkat & Richert, Halberstam & Richert,
and Nathanson.
Suppose A is a finite sequence of integers and P is a set of primes. Write Ad for the number of items in A that are divisible by d, and write P for the product of the elements in P that are less than z. Write ω for a multiplicative function such that ω/p is approximately the proportion of elements of A divisible by p, write X for any convenient approximation to |A|, and write the remainder as
Write S for the number of items in A that are relatively prime to P. Write
Write ν for the number of distinct prime divisors of m. Write F1 and f1 for functions satisfying certain difference differential equations.
We assume the dimension is 1: that is, there is a constant C such that for 2 ≤ z < w we have
Then the Jurkat–Richert theorem states that for any numbers y and z with 2 ≤ z ≤ y ≤ X we have
and