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

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

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 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

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

(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

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

and

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

Notes

1. Nathanson, Melvyn B. (1996). Additive Number Theory: The Classical Bases. Graduate Texts in Mathematics 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 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 4. London: Academic Press. ISBN 0-12-318250-6. MR 54:12689. Zbl 0298.10026.