Let $P_k$ be the set of integers with at most $k$ prime factors (counting with multiplicity, say). There is an almost-prime number theorem which gives asymptotic estimates of the size of $P_k$, and (presumably) one can formulate a question analogous to the Riemann Hypothesis on the size of the corresponding error term. My question is: what are the best known unconditional estimates? In particular, if $k$ is large (say $100$) can we obtain a significantly better error term than we know for $P_1$ (say: $n^{1-\varepsilon}$)?

I am also interested in the same question if $P_k$ is replaced by $P'_k$, the set of integers $n$ with at most $k$ prime factors (with multiplicity) none of which is smaller than $\log n$. (Here I am not too concerned about the precise growth rate $\log n$; larger is presumably unhelpful, but $\log\log n$ or $\log_* n$ would be fine)

I would especially be interested in answers of the form 'this question is likely hard because...'.

EDIT: The answer I would like is something of the form $|P_k(n)|=m_k(n)+e_k(n)$ where $m_k(n)$ is the `main term' and $e_k(n)$ the error, and the property I would like is not that $m_k(n)$ should be the simplest function which is $(1+o(1))|P_k(n)|$, but rather $m_k(n)$ should be whatever function with nice analytic properties allows us to make $e_k(n)$ small. For the kinds of things I want to do, I wouldn't need an explicit formula for $m_k(n)$; the approximation which exists (see answers below) and 'niceness' is enough. Let me not try to say exactly what 'nice' should be, since it's not too critical; certainly it should allow us to show (by the obvious argument) that members of $P_k$ have about the same density on the interval $[n,2n]$ as in any sub-interval of length at least $n^{\theta+\varepsilon}$ if we are given $e_k(n)=O(n^\theta)$, for any $\varepsilon>0$ and sufficiently large $n$.

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