claimroutineinferred_dep

Uniform upper sieve on affine fibers

Bounds lattice points on an affine fiber with prime values by

X^2/(\log X)^{t-1}

uniform count of bad configurations

Claim·§6.6 A uniform sieve on each affine fiber·inside lemPrime patterns with controlled reduction orders

Statement (paper)

For each form index

\nu

and each prime value

r\asymp_A X

of the

\nu

th form, the number of

T\in X\mathcal B\cap\Z^3

with

F_\nu(T)=\epsilon_\nu k_\nu r

for which all other

\Psi_j(T)

are primes is

\ll_A X^2/(\log X)^{t-1}

, uniformly in

r

and independently of

L

In plain English

Fix a form index

\nu

and a prime value

r\asymp_A X

of the

\nu

th form. The number of

T\in X\mathcal B\cap\Z^3

with

F_\nu(T)=\epsilon_\nu k_\nu r

for which all the other

\Psi_j(T)

are prime is

\ll_A X^2/(\log X)^{t-1}

, uniformly in

r

and independently of

L

Why it's here

The condition

F_\nu(T)=\epsilon_\nu k_\nu r

cuts out a two-dimensional affine fibre of lattice points. The upper sieve (Lemma 6.3) is applied on that fibre to the other

t-1

forms, and the lower bound on

G(Z)

turns its main term into

X^2/(\log X)^{t-1}

. The bound is uniform in

r

and independent of

L

, so the later claims can sum it over prime values

r

to bound the number of bad configurations without any dependence on

L

Classified routine:

Direct application of the paper's elementary upper sieve (Lemma 6.3) on a two-dimensional affine fibre, combined with the lower bound for G(Z) from the earlier claim; this is a standard Selberg/Brun-type upper-bound sieve argument.

Uses

lemElementary upper sieve with uniform errorsclaimSieve product lower bound G(Z)

Used by

claimSurviving prime pattern satisfies order conditions

Sources

PaperOpen

Section: §6.6 A uniform sieve on each affine fiber
Page: —
Label: —
Line: sections/06-primes.tex:615

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Uniform upper sieve on affine fibers" rests on pri:upper-sieve
info
inferred_dep
outline: "Uniform upper sieve on affine fibers" rests on claim-pri-pattern-4