lemmaroutine

Elementary upper sieve with uniform errors

Bounds lattice points avoiding residue sets by n2/G(Z)+O(nZ6+Z8)n^2/G(Z)+O(nZ^6+Z^8)
sieve bound for counting bad prime values
Lemma·pri:upper-sieve·§6.4 An elementary upper sieve with uniform errors

Statement (paper)

The number of points in this square avoiding every Ωp\Omega_p is at most n2G(Z)+O(nZ6+Z8),G(Z)=∑e≤Z squarefreep∣e⇒p∈PZ∏p∣eg(p)1−g(p). \frac{n^2}{G(Z)}+O(nZ^6+Z^8),\qquad G(Z)=\sum_{\substack{e\le Z\text{ squarefree}\\p\mid e\Rightarrow p\in\mathcal P_Z}} \prod_{p\mid e}\frac{g(p)}{1-g(p)}. The error constant is absolute.

In plain English

In a square of side nn, the number of lattice points that avoid every excluded residue set Ωp\Omega_p is at most n2/G(Z)+O(nZ6+Z8)n^2/G(Z)+O(nZ^6+Z^8), where G(Z)G(Z) sums ∏p∣eg(p)/(1−g(p))\prod_{p\mid e}g(p)/(1-g(p)) over squarefree e≤Ze\le Z with all prime factors in PZ\mathcal P_Z. The implied constant is absolute.

Why it's here

This is the sieve input for the later counting claims: it gives an upper bound on points avoiding the bad residue classes, with an explicit error that is uniform in the data. The main term n2/G(Z)n^2/G(Z) has the size of a Selberg-type sieve bound, and the error O(nZ6+Z8)O(nZ^6+Z^8) stays small when ZZ is a small power of nn. That is what the claims on bad prime values and on affine fibers need to control their counts uniformly.
Classified routine:
This is the Selberg upper-bound sieve in two dimensions with main term n^2/G(Z), the classical Selberg sieve as in Halberstam–Richert, *Sieve Methods*. The explicit error O(nZ^6+Z^8) comes from routine bookkeeping of lattice-point counts in residue classes.

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
Page
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Label
pri:upper-sieve
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sections/06-primes.tex:458
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