On almost-prime $k$-tuples

Bin Chen

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arXiv · 2301.05044

On almost-prime $k$-tuples

Bin Chen

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Abstract

Let $\tau$ denote the divisor function and $\mathcal{H}=\{h_{1},...,h_{k}\}$ be an admissible set. We prove that there are infinitely many $n$ for which the product $\prod_{i=1}^{k}(n+h_{i})$ is square-free and $\sum_{i=1}^{k}\tau(n+h_{i})\leq \lfloor \rho_{k}\rfloor$, where $\rho_{k}$ is asymptotic to $\frac{2126}{2853} k^{2}$. It improves a previous result of M. Ram Murty and A. Vatwani, replacing $2126/2853$ by $3/4$. The main ingredients in our proof are the higher rank Selberg sieve and Irving-Wu-Xi estimate for the divisor function in arithmetic progressions to smooth moduli.

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Bin Chen. 2023-01-12. On almost-prime $k$-tuples. https://arxiv.org/abs/2301.05044

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