Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Variants of the Selberg sieve, and almost prime k-tuples (2203.09432v1)

Published 17 Mar 2022 in math.NT

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.

Summary

We haven't generated a summary for this paper yet.