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

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

Notes

1. Nathanson, Melvyn B. (1996). Additive Number Theory: The Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 978-0-387-94656-6. Zbl 0859.11003. Retrieved 2009-03-14.
2. Jurkat, W. B.; Richert, H.-E. (1965). "An improvement of Selberg's sieve method I" (PDF). Acta Arithmetica. XI: 217–240. ISSN 0065-1036. Zbl 0128.26902. Retrieved 2009-02-17.
3. Diamond, Harold G.; Halberstam, Heini (2008). A Higher-Dimensional Sieve Method: with Procedures for Computing Sieve Functions. Cambridge Tracts in Mathematics. Vol. 177. With William F. Galway. Cambridge: Cambridge University Press. ISBN 978-0-521-89487-6. Zbl 1207.11099.
4. Halberstam, Heini; Richert, H.-E. (1974). Sieve Methods. London Mathematical Society Monographs. Vol. 4. London: Academic Press. ISBN 0-12-318250-6. MR 0424730. Zbl 0298.10026.