A1 Refereed original research article in a scientific journal
Weighted sieves with switching
Authors: Matomäki, Kaisa; Zuniga-Alterman, Sebastian
Publisher: Cambridge University Press (CUP)
Publication year: 2025
Journal: Mathematical Proceedings of the Cambridge Philosophical Society
Journal name in source: Mathematical Proceedings of the Cambridge Philosophical Society
First page : 1
Last page: 22
ISSN: 0305-0041
eISSN: 1469-8064
DOI: https://doi.org/10.1017/S0305004125000258
Web address : https://doi.org/10.1017/s0305004125000258
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/498941075
Weighted sieves are used to detect numbers with at most S prime factors with S∈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 utilise 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,P3) when both the original and the switched problem have level of distribution at least 0.267
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Funding information in the publication:
The authors would like to thank the anonymous referee for very careful reading of the paper. Both authors were supported by the Finnish Centre of Excellence in Randomness and Structures (Research Council of Finland grants no. 346307 and 364214), and the first author was additionally supported by Research Council of Finland grant no. 333707.
