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 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

$r_A(d)=\left|A_d\right|-\frac{\omega (d)}{d}X.$

Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write

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

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 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

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

(The book of Diamond & Halberstam 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

$S(A,P,z)\le XV(z)(F_1\left(\frac{\log y}{\log z}\right)+O\left(\frac{(\log \log y{)}^{3/4}}{(\log y{)}^{1/4}}\right))+\sum _{m|P(z),m<y}{4}^{\nu (m)}|r_A(m)|$

and

$S(A,P,z)\ge XV(z)(f_1\left(\frac{\log y}{\log z}\right)-O\left(\frac{(\log \log y{)}^{3/4}}{(\log y{)}^{1/4}}\right))-\sum _{m|P(z),m<y}{4}^{\nu (m)}|r_A(m)|.$