Second-order terms for the variances and covariances of the number of prime factors—Including the square free case

We obtain second-order terms for the variance and covariance of $\text{Ω(n)}$ and ω(n), the number of prime divisors counted with and without multiplicity, and connect these results to a formula of Renyi. We discuss the heuristic connection with the Landau-Sathe extension of the prime number theorem and develop new expansions for the mean and variance of ω(n) in the square free case.