论文标题
Selberg筛子的变体，几乎是Prime K-tuples
Variants of the Selberg sieve, and almost prime k-tuples
论文作者
论文摘要
令$ k \ geq 2 $和$ \ mathcal {p}（n）=（a_1 n + b_1）\ cdots（a_k n + b_k）$，其中所有$ a_i，b_i $都是整数。假设$ \ Mathcal {p}（n）$没有固定的prime除数。对于每种选择$ k $，都知道存在整数$ \ varrho_k $，使得$ \ mathcal {p}（n）$最多有$ \ varrho_k $ prime prime prime因素经常经常。我们使用了一种新的加权筛设置与一种称为$ \ varepsilon $ -trick的设备，以提高$ k \ geq 7 $的$ \ varrho_k $的可能值。作为我们方法的副产品，我们提高了$ k \ geq 4 $的$ \ varrho_k $的条件可能值，假设是普遍的Elliott-Halberstam猜想。
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.
