claimroutine

Sieve product lower bound G(Z)

Bounds sieve sum G(Z)≫(log⁡X)sG(Z)\gg(\log X)^s, or ≫log⁡X\gg\log X in the 1 mod 4 case
lower bound needed by the upper sieve
Claim·§6.4 An elementary upper sieve with uniform errors·inside lemPrime patterns with controlled reduction orders

Statement (paper)

Set Z=X1/100Z=X^{1/100}. If PZ\mathcal P_Z contains all primes above a fixed threshold up to ZZ, and g(p)=s/p+Os(p−2)g(p)=s/p+O_s(p^{-2}) for a fixed positive integer ss, then G(Z)≫(log⁡X)sG(Z)\gg(\log X)^s. If instead PZ\mathcal P_Z contains the primes p≡1(mod4)p\equiv1\pmod4 above a fixed threshold up to ZZ, with at most one additional prime omitted, and g(p)=2/p+O(p−2)g(p)=2/p+O(p^{-2}), then G(Z)≫log⁡XG(Z)\gg\log X, uniformly in the omitted prime. The implicit constants may depend on the fixed threshold, the fixed density estimates and ss, but not on XX or the omitted prime.

In plain English

With Z=X1/100Z=X^{1/100}, the sieve sum G(Z)G(Z) is at least a constant times (log⁡X)s(\log X)^s when PZ\mathcal P_Z contains all primes in a range above a fixed threshold up to ZZ and g(p)=s/p+Os(p−2)g(p)=s/p+O_s(p^{-2}). If instead PZ\mathcal P_Z contains the primes p≡1(mod4)p\equiv1\pmod4 above the threshold up to ZZ, except possibly one omitted prime, and g(p)=2/p+O(p−2)g(p)=2/p+O(p^{-2}), then G(Z)≫log⁡XG(Z)\gg\log X, uniformly in the omitted prime.

Why it's here

This gives the lower bound on G(Z)G(Z) that an upper-bound sieve needs, so that the sieve's main term, which is of size 1/G(Z)1/G(Z), is small enough to give the right upper bounds. The idea is that the product over primes of 1+g(p)/(1−g(p))1+g(p)/(1-g(p)) behaves like ∏(1−1/p)−s\prod(1-1/p)^{-s}, which is of size (log⁡Z)s≍(log⁡X)s(\log Z)^s\asymp(\log X)^s. In the p≡1(mod4)p\equiv1\pmod4 case the same argument applies with density 2/p2/p over a set of primes of relative density 1/21/2, and dropping one prime changes the product only by a bounded factor. Both bounds feed the uniform upper-sieve estimates in the later claims.
Classified routine:
Standard Selberg-sieve estimate of G(Z) from below by the Euler product of (1-g(p))^{-1}, evaluated with Mertens' theorem and its version for primes congruent to 1 mod 4 (prime number theorem in arithmetic progressions); omitting one prime changes the product by only a bounded factor.

Uses

Nothing else in the paper

Used by

claimFew primes with large common order factorclaimUniform upper sieve on affine fibers

Sources

PaperOpen
Section
§6.4 An elementary upper sieve with uniform errors
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