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 A_d 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

${\displaystyle V(z)=\prod _{p\in P,p<z}\left(1-{\frac {\omega (p)}{p}}\right).}$

Write ν(m) for the number of distinct prime divisors of m. Write F₁ and f₁ for functions satisfying certain difference differential equations (see Diamond & Halberstam^{[3]: 67–68} for the definition and properties).

We assume the dimension (sifting density) is 1: that is, there is a constant C such that for 2 ≤ z < w we have

${\displaystyle \prod _{z\leq p<w}\left(1-{\frac {\omega (p)}{p}}\right)^{-1}\leq \left({\frac {\log w}{\log z}}\right)\left(1+{\frac {C}{\log z}}\right).}$

(The book of Diamond & Halberstam^[3] extends the theorem to dimensions higher than 1.) Then the Jurkat–Richert theorem states that for any numbers y and z with 2 ≤ z ≤ y ≤ X we have

${\displaystyle S(A,P,z)\leq XV(z)\left(F_{1}\left({\frac {\log y}{\log z}}\right)+O\left({\frac {(\log \log y)^{3/4}}{(\log y)^{1/4}}}\right)\right)+\sum _{m|P(z),m<y}4^{\nu (m)}\left|r_{A}(m)\right|}$

and

${\displaystyle S(A,P,z)\geq XV(z)\left(f_{1}\left({\frac {\log y}{\log z}}\right)-O\left({\frac {(\log \log y)^{3/4}}{(\log y)^{1/4}}}\right)\right)-\sum _{m|P(z),m<y}4^{\nu (m)}\left|r_{A}(m)\right|.}$