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arXiv · 2207.05038

Almost primes in almost all short intervals II

Kaisa Matomäki·Joni Teräväinen
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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.

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Kaisa Matomäki, Joni Teräväinen. 2023-02-01. Almost primes in almost all short intervals II. https://doi.org/10.1090/tran%2F8869

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