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arXiv · 2405.19063

Weighted sieves with switching

Kaisa Matomäki·Sebastian Zuniga Alterman
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Abstract

Weighted sieves are used to detect numbers with at most $S$ prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes $p$ such that $p+2$ has at most two prime factors), one can utilize the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes. As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

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Kaisa Matomäki, Sebastian Zuniga Alterman. 2025-03-14. Weighted sieves with switching. https://doi.org/10.1017/s0305004125000258

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