title: Jurkat–Richert theorem
text: # 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 $conjecture.^{:272}$ It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.
  
  ## Statement of the theorem
  
  This formulation is from Diamond & $Halberstam.^{:81}$ Other formulations are in Jurkat & $Richert,^{:230}$ Halberstam & $Richert,^{:231}$ and $Nathanson.^{:257}$
  
  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
  
      $r_{A}(d)=\left|A_{d}\right|-{\frac {\omega (d)}{d}}X.$
  
  Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write
  
      $V(z)=\prod _{p\in P,p<z}\left(1-{\frac {\omega (p)}{p}}\right).$
  
  Write ν(m) for the number of distinct prime divisors of m. Write $F_1$ and $f_1$ for functions satisfying certain difference differential equations (see Diamond & $Halberstam^{:67–68}$ 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
  
      $\prod _{z\leq p<w}\left(1-{\frac {\omega (p)}{p}}\right)^{-1}\leq \left({\frac {\log w}{\log z}}\right)\left(1+{\frac {C}{\log z}}\right).$
  
  (The book of Diamond & Halberstam 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
  
      $S(A,P,z)\leq XV(z)\left(F_{1}\left({\frac {\log y}{\log z}}\right)+O\left({\frac {(\log \log y)^{3/4}}{(\log y)^{1/4}}}\right)\right)+\sum _{m|P(z),m<y}4^{\nu (m)}\left|r_{A}(m)\right|$
  
  and
  
      $S(A,P,z)\geq XV(z)\left(f_{1}\left({\frac {\log y}{\log z}}\right)-O\left({\frac {(\log \log y)^{3/4}}{(\log y)^{1/4}}}\right)\right)-\sum _{m|P(z),m<y}4^{\nu (m)}\left|r_{A}(m)\right|.$
  