= Almost-primes from an upper-bound sieve and Buchstab identity
Let $F(n)=\prod_{j=1}^{k}(a_jn+b_j)$ be a product of fixed pairwise nonproportional <linear polynomials> with positive <integer> leading coefficients. Assume admissibility: no <prime number> divides $F(n)$ for every <integer> $n$. There are constants $\delta,c>0$ such that, for all large $X$, at least $cX/(\log X)^k$ values $1\le n\le X$ have no <prime factor> of $F(n)$ below $X^\delta$. These are <almost prime> values with a bounded total number of <prime factors>, counted with <multiplicity>.
A rigorous positive count needs a <lower-bound sieve> input, obtainable from the <fundamental lemma of sieve theory>. If $\rho_F(d)$ counts roots modulo squarefree $d$, then the <polynomial root density in a sieve> gives $X\rho_F(d)/d+O(\rho_F(d))$. Take distribution level $D=X^{1/2}$ and sifting limit $z=X^\delta$. The total remainder up to $D$ is $O_k(D(\log(2D))^k)$, while <Mertens' theorems> give $\prod_{p<z}(1-\rho_F(p)/p)\asymp_F(\log z)^{-k}$. Choosing $\delta$ sufficiently small makes the main-term error in the <fundamental lemma of sieve theory> less than half the main term, proving the lower count.<Some sieve theory by Terence Tao (254A Notes 4)>[Corollary 19; equation (26) and Corollary 21 illustrate the two-linear-form case]{ref}
Since $|F(n)|\ll_F X^k$, each surviving nonzero value has at most $k/\delta+O_F(1)$ <prime factors>, with <multiplicity>. The <Buchstab identity> organizes the removal of successive <prime factors>, and <upper-bound sieves> control removed terms; merely observing convergence of a prime tail does not establish that its subtraction leaves a positive main term.
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