On the middle divisors of an integer

Abstract. Given a positive integer \(n\), let \(\rho_{1}(n)= \max\{d\ |\ n : d\leq \sqrt{n}\}\) and \(\rho_{2}(n)= \min\{d\ |\ n : d\ge \sqrt{n}\}\) stand for the middle divisors of \(n\). We obtain improvements and new estimates for sums involving these two functions.