Almost primes in almost all short intervals II
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- by Kaisa Matomäki and Joni Teräväinen
- Trans. Amer. Math. Soc. 376 (2023), 5433-5459
- DOI: https://doi.org/10.1090/tran/8869
- Published electronically: May 9, 2023
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Abstract:
We show that, for almost all $x$, the interval $(x, x+(\log x)^{2.1}]$ contains products of exactly two primes. This improves on a work of the second author that had $3.51$ in place of $2.1$. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials.References
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Bibliographic Information
- Kaisa Matomäki
- Affiliation: Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
- Email: ksmato@utu.fi
- Joni Teräväinen
- Affiliation: Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
- MR Author ID: 1171846
- Email: joni.p.teravainen@gmail.com
- Received by editor(s): August 5, 2022
- Received by editor(s) in revised form: November 2, 2022
- Published electronically: May 9, 2023
- Additional Notes: The first author was supported by Academy of Finland grant no. 285894. The second author was supported by Academy of Finland grant no. 340098.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 5433-5459
- MSC (2020): Primary 11N05, 11N36
- DOI: https://doi.org/10.1090/tran/8869
- MathSciNet review: 4630750