On almost-prime $k$-tuples

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 ${\sum }_{i=1}^k\tau (n+h_i)\le \lfloor {\rho }_k\rfloor$, 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.