Bounds on the counting function for almost-primes

Question

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 $n$.

Answer 1

The typical asymptotic formula, for fixed $k$, is $$ \#\{n\le x\colon \Omega(n)=k\} = \frac{x(\log\log x)^{k-1}}{(k-1)!\log x} + O_k \bigg( \frac{x(\log\log x)^{k-2}}{\log x} \bigg). $$ In this form, the error term is likely to be best possible! Presumably the "correct" main term would have the form $$ \frac x{\log x} P_k(\log\log x), $$ where $P_k(t)$ is a polynomial whose leading term is $\frac{t^{k-1}}{(k-1)!}$. Only in this form would we potentially be able to save a power of $x$ in the error term.

Check out Tenenbaum's book (the chapter on Selberg–Delange) to see if this application is in there, or if it can be derived from that method, perhaps using the function $\Lambda_k$.

Answer 2

Self-answer, since the answer given addresses the question I didn't mean to ask - I would appreciate it if experts tell me whether or not my attempts at finding out by google and guesstimated calculations make sense.

The situation seems to be:

It is believable (depending on how much you trust Cramer-type models) that the decomposition I would like actually holds even with $e_k(n)$ growing poly-logarithmically (for any $k$ including $1$, so counting the primes). For many practical purposes this is much stronger than the Riemann Hypothesis, but it's actually consistent with the Riemann Hypothesis failing (?), even quite badly, as that conjecture would make some statement stronger than 'nice analytic properties' about the main term.

Probably the equivalent to the Riemann Hypothesis for almost-primes (i.e. proving we can obtain $|e_k(n)|=O(n^{1/2+\varepsilon})$ for each $\varepsilon>0$ is easier than for primes, but 'easier than something very very hard' is not too helpful.

As to unconditional results, nothing seems to have been worked out but standard methods should give results for $|P_k(n)|$ of comparable strength to $|P_1(n)|$ (?).

Where one does gain something is if one asks the corresponding question (correctly phrased, which probably means counting with weight and looking for a main term of order $n$) with respect to a non-trivial Dirichlet character, as then the nastiness caused by potential Siegel zeroes goes away, so for example the counting function for almost-primes in arithmetic progressions has effective error bounds (?), even when the common difference is polylogarithmic in $n$.