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
Suppose A is a finite sequence of integers and P is a set of primes. Write Ad 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
Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write
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: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
(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
- 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.
- 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.
- 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.
- 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.