On almost-prime k-tuples

Let $\tau$ denote the divisor function and ${\mathscr{H}}=\{h_1,\dots ,h_k\}$ 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 Ram Murty and Vatwani, replacing $3/4$ by $2126/2853$. 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.