  The paper is concerned with estimating the number of integers smaller than
$x$ whose largest prime divisor is smaller than $y$, denoted $\psi (x,y)$. Much
of the related literature is concerned with approximating $\psi (x,y)$ by
Dickman's function $\rho (u)$, where $u=\ln x/\ln y$. A typical such result is
that $$ \psi (x,y)=x\rho (u)(1+o(1)) \eqno (1) $$ in a certain domain of the
parameters $x$ and $y$.
  In this paper a different type of approximation of $\psi (x,y)$, using
iterated logarithms of $x$ and $y$, is presented. We establish that $$ \ln
(\frac {\psi}{x})=-u [\ln ^{(2)}x-\ln ^{(2)}y+\ln ^{(3)}x-\ln ^{(3)}y+\ln
^{(4)}x-a] \eqno (2) $$ where $\underbar{a}<a<\bar{a}$ for some constants
$\underbar{a}$ and $\bar{a}$ (denoting by $\ln ^{(k)}x=\ln ...\ln x$ the
$k$-fold iterated logarithm). The approximation (2) holds in a domain which is
complementary to the one on which the approximation (1) is known to be valid.
One consequence of (2) is an asymptotic expression for Dickman's function,
which is of the form $\ln \rho (u)=-u[\ln u+\ln ^{(2)}u](1+o(1))$, improving
known asymptotic approximations of this type. We employ (2) to establish a
version of Bertrand's Conjecture, and indicate how this method may be used to
sharpen the result.
