Variants of the Selberg sieve, and almost prime k-tuples

TL;DR

This paper advances sieve methods to better understand the frequency of almost prime values of polynomial products, improving bounds on the number of prime factors for such polynomials using new techniques and conjectural assumptions.

Contribution

It introduces a new weighted sieve setup and an -trick to improve bounds on prime factors of polynomial products, especially for larger degrees.

Findings

Improved bounds on _k for krom 7 using new sieve techniques.

Conditional improvements on _k for krom 4 assuming Elliott--Halberstam conjecture.

Enhanced understanding of almost prime k-tuples in polynomial sequences.

Abstract

Let $k\geq 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$ where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no fixed prime divisors. For each choice of $k$ it is known that there exists an integer $\varrho_k$ such that $\mathcal{P} (n)$ has at most $\varrho_k$ prime factors infinitely often. We used a new weighted sieve set-up combined with a device called an $\varepsilon$-trick to improve the possible values of $\varrho_k$ for $k\geq 7$. As a by-product of our approach, we improve the conditional possible values of $\varrho_k$ for $k\geq 4$, assuming the generalized Elliott--Halberstam conjecture.