Almost-prime $k$-tuples

TL;DR

This paper improves bounds on the number of prime factors in almost-prime $k$-tuples using a novel weighted sieve method, advancing understanding of their prime factorization properties.

Contribution

It introduces a new weighted sieve technique that reduces the upper bounds on prime factors for almost-prime $k$-tuples when $k\u2265 4$.

Findings

Improved bounds on the number of prime factors in almost-prime $k$-tuples.

Demonstrated effectiveness of the new sieve method for $k\u2265 4$.

Enhanced understanding of the distribution of prime factors in polynomial products.

Abstract

Let $k\ge 2$ and $\Pi(n)=\prod_{i=1}^k(a_in+b_i)$ for some integers $a_i, b_i$ ($1\le i\le k$). Suppose that $\Pi(n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $\Omega(\Pi(n))\le r_k$ holds for some integer $r_k$ which is asymptotic to $k\log{k}$. We use a new kind of weighted sieve to improve the possible values of $r_k$ when $k\ge 4$.