Reference request for proof of Landau's generalised PNT Could someone please point me in the direction of a proof for Landau's asymptotic formula for k-almost primes: $$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$ I realise that it was derived directly from the PNT - would like to see the steps involved though.

The original source, as far as I know, is Landau's _Handbuch der Lehre von der Verteilung der Primzahlen_. An approachable modern version (in English!) is

Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press (1995).