ON ULTRAFRIABLE INTEGERS

Say that an integer n is y-ultrafriable if its canonical decomposition is free of prime powers exceeding y. We investigate the asymptotic behaviour of the distribution function Ƴ(x, y), equal to the number of y-ultrafriable integers not exceeding x. The study being restricted to the range ψ(y)>2 log x (where ψ denotes Chebyshev's function) by a symmetry argument, and writing Ψ(x, y) for the number of y-friable integers not exceeding x, β=β(x, y) for the saddle point associated to the Dirichlet series $Z(s,y):=\prod_{p\leqslant y}(1-p^{-(\nu_{p}+1) s})/( 1-p^{-s})$ where $\nu_{p}:=\lfloor(\log y)/\log p\rfloor$, we obtain full description via a number of effective estimates, qualitative versions of which may be stated as follows:$\Upsilon(x, y)\sim\Psi(x, y)\Leftrightarrow y/(\log x)^{2}\to \infty\ (x\to \infty)$;$\Upsilon(x, y) = o(\Psi(x, y))\Leftrightarrow y/(\log x)^{2}\to 0\ (x\to \infty)$;$\Upsilon(x, y)\sim x^{\beta} Z(\beta, y)G(\beta \sqrt{\sigma_{2}})\ (x\to \infty, 2 \log x < \psi(y)\leqslant (\log x)^{3})$.Here $\sigma_{2}:=\hbox{d}^{2}\{\log Z(\beta, y)\}/\hbox{d} s^{2}$ and $G(z):=\hbox{e}^{z^{2}/2}\int_{z}^{\infty} \hbox{e}^{-t^{2}/2}\hbox{d}t/\sqrt{2\pi}$ (z>0). Moreover, the phase transition between domains (i) and (ii) is quantitatively described in terms of the Dickman function, the Gaussian behaviour as log x approaches $\frac{1}{2}\psi(y)$ is made explicit, and a quantitative estimate for the local behaviour with respect to the variable x is derived.