claimroutineinferred_dep

Green-Tao-Ziegler lower bound for prime patterns

Counts prime patterns, at least

c_A(\log L)^{-C_0}X^3/(\log X)^t

many prime patterns via Green–Tao–Ziegler

Claim·§6.3 A lower bound for the prime patterns·inside lemPrime patterns with controlled reduction orders

Statement (paper)

For sufficiently large

X

, depending on

A

and

L

, the number

N_{A,L}(X)

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

for which every

\Psi_\nu(T)

is prime and all imposed congruences hold satisfies

N_{A,L}(X)\ge c_A(\log L)^{-C_0}\frac{X^3}{(\log X)^t},

with

c_A>0

independent of

L

In plain English

For large

X

(depending on

A

and

L

), the number

N_{A,L}(X)

of integer points

T

X\mathcal B

with every

\Psi_\nu(T)

prime and all imposed congruences satisfied is at least

c_A(\log L)^{-C_0}X^3/(\log X)^t

, where

c_A>0

does not depend on

L

Why it's here

This applies the Green–Tao–Ziegler count for linear forms in primes to the thirteen forms

\Psi_\nu

and the congruence class set up in the previous claim. It guarantees many prime patterns, with the

L

-dependence only a mild

(\log L)^{-C_0}

loss, so that later steps can discard those patterns with bad common odd factors above

L

and still have one left over.

Classified routine:

A direct application of the Green–Tao–Ziegler asymptotic for linear equations in primes (Green–Tao 2010, with Green–Tao–Ziegler inverse theorem 2012) to the thirteen forms and the residue class from the previous claim. The only extra work is bookkeeping of the local densities to get the (log L)^{-C_0} dependence.

Uses

claimFinite multiplier set and initial residue class

Used by

claimSurviving prime pattern satisfies order conditions

Sources

PaperOpen

Section: §6.3 A lower bound for the prime patterns
Page: —
Label: —
Line: sections/06-primes.tex:318

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Green-Tao-Ziegler lower bound for prime patterns" rests on claim-pri-pattern-2