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If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is pretty big, and if $k$ is pretty big, and if $\ln \ln x - \ln \ln y$ is pretty close to $k$, then I'm fairly sure that $$\Phi_k(x, y) \approx \frac{e^{-\gamma}x }{\ln x} \frac{(\ln \ln x - \ln \ln y)^{k-1}}{(k-1)!}$$ with $\gamma$ the Euler-Mascheroni constant.

This result seems almost exercise-level basic and, in the case that $\ln \ln y = o(\sqrt{\ln \ln x})$, is given as a literal exercise in a paper of Tenenbaum. However, my current approach to proving it for larger $y$ involves taking a detour through this paper to prove, for any positive real $\alpha$, $$\lim_{k \rightarrow \infty}\,\, \frac{1}{(\alpha k)^k} \cdot\int_{\substack{x_1, \dots, x_k \ge 1\\ \sum x_i \le e^{\alpha k}}} \prod_{i \le k} \frac{dx_i}{x_i} \approx \frac{e^{-\gamma/\alpha}}{\Gamma\big(1 + \frac{1}{\alpha}\big)}.$$

This seems a bit like overkill to me. Does anyone know of a good source for asymptotics/error on $\Phi_k(x, y)$?

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Asymptotics of such integral may be obtained by applying appropriate Tauberian theorems. If we denote $x_i=e^{t_i}$, then your integral is $F(e^{\alpha k})$, where $F(T)$ is a volume of the set $\{t_i\geqslant 0,\sum_{i=1}^k e^{t_i}\leqslant T\}$. For getting some information of $F(T)$ we take some $\lambda>0$ and integrate $\exp(-\lambda\sum e^{t_i})$ over $[0,\infty)^k$. At first, this integral factorizes and equals $(\int_0^\infty \exp(-\lambda e^t)dt)^k=(-Ei(-\lambda))^k$. On the other hand, it equals $$ \int_0^{\infty}\mu\{\exp(-\lambda\sum e^{t_i})\geqslant x\}dx= \int_0^{\infty}\mu\{\exp(-\lambda\sum e^{t_i})\geqslant e^{-s}\}e^{-s}ds=\\ \int_0^\infty F(s/\lambda)e^{-s}ds. $$ Asymptotics of $F(T)$ for large $T$ should be defined by asymptotics of $\int_0^\infty F(s/\lambda)e^{-s}ds$ for large $\lambda$ under some assumptions like monotonicity. (Not finished yet.)

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