Counting integers with k large prime divisors

Question

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 $\Phi_k(x, y)$?

Answer 1

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.)

Answer 2

In case anyone else is looking for this result, as I was, this is treated in Alladi's paper, "The distribution of ν(n) in the sieve of Eratosthenes", Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 130, 129–148.

In particular, it is shown there that if $\exp((\log_2 x)^3)<y<\sqrt{x}$, $u=\frac{\log x}{\log y}$ and $\xi = \frac{k}{\log u -\gamma}$ and $r>0$ is fixed, then uniformly for $1\leq k <r \log u$,

$$\Phi_k(x,y) = \frac{xe^{-\gamma\xi}}{\log x \ \Gamma(1+\xi)} \cdot \frac{(\log u)^{k-1}}{(k-1)!}\left(1+O_r\left(\frac{1}{\sqrt{\log u}}\right)\right).$$

The conjecture above follows since if $\log \log x-\log \log y \approx k$ then $\xi \approx 1$.

See also Balazard, Michel, "Unimodalité de la distribution du nombre de diviseurs premiers d'un entier", Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 255–270. Where a slightly more cumbersome version is given that is applicable in a greater range of $u$.