Jurkat–Richert theorem

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.[1]: 272  It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.[2] Statement of the theorem This formulation is from Diamond & Halberstam.[3]: 81  Other formulations are in Jurkat & Richert,[2]: 230  Halberstam & Richert,[4]: 231  and Nathanson.[1]: 257  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(z) for the product of the elements in P that are less than z. Write ω(d) for a multiplicative function such that ω(p)/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 {displaystyle r_{A}(d)=left|A_{d}right|-{frac {omega (d)}{d}}X.} Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write {displaystyle V(z)=prod _{pin P,p

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