On almost-prime k-tuples

Abstract

Let [Formula: see text] denote the divisor function and [Formula: see text] be an admissible set. We prove that there are infinitely many [Formula: see text] for which the product [Formula: see text] is square-free and [Formula: see text], where [Formula: see text] is asymptotic to [Formula: see text]. It improves a previous result of Ram Murty and Vatwani, replacing [Formula: see text] by [Formula: see text]. 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.