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Almost primes in almost all short intervals II
- Source :
- Trans. Amer. Math. Soc. 376, 5433-5459, 2023
- Publication Year :
- 2022
-
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.<br />Comment: 26 pages; referee comments incorporated; to appear in Trans. Am. Math. Soc
- Subjects :
- Mathematics - Number Theory
11N05, 11N36
Subjects
Details
- Database :
- arXiv
- Journal :
- Trans. Amer. Math. Soc. 376, 5433-5459, 2023
- Publication Type :
- Report
- Accession number :
- edsarx.2207.05038
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1090/tran/8869