Jurkat–Richert theorem - Alchetron, The Free Social Encyclopedia

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 ) = | A d | − ω ( 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 ) = ∏ p ∈ P , p < z ( 1 − ω ( 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

∏ z ≤ p < w ( 1 − ω ( p ) p ) − 1 ≤ ( log ⁡ w log ⁡ z ) ( 1 + 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 ) ≤ X V ( z ) ( F 1 ( log ⁡ y log ⁡ z ) + O ( ( log ⁡ log ⁡ y ) 3 / 4 ( log ⁡ y ) 1 / 4 ) ) + ∑ m | P ( z ) , m < y 4 ν ( m ) | r A ( m ) |

and

S ( A , P , z ) ≥ X V ( z ) ( f 1 ( log ⁡ y log ⁡ z ) − O ( ( log ⁡ log ⁡ y ) 3 / 4 ( log ⁡ y ) 1 / 4 ) ) − ∑ m | P ( z ) , m < y 4 ν ( m ) | r A ( m ) | .

References

Jurkat–Richert theorem Wikipedia

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