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The paper is concerned with estimating the number of integers smaller than$x$ whose largest prime divisor is smaller than $y$, denoted $\psi (x,y)$. Muchof the related literature is concerned with approximating $\psi (x,y)$ byDickman's function $\rho (u)$, where $u=\ln x/\ln y$. A typical such result isthat $$ \psi (x,y)=x\rho (u)(1+o(1)) \eqno (1) $$ in a certain domain of theparameters $x$ and $y$.  In this paper a different type of approximation of $\psi (x,y)$, usingiterated 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 iscomplementary 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 $\ln \rho (u)=-u[\ln u+\ln ^{(2)}u](1+o(1))$, improvingknown asymptotic approximations of this type. We employ (2) to establish aversion of Bertrand's Conjecture, and indicate how this method may be used tosharpen the result.

Iterated logarithm approximations to the distribution of the largest prime divisor