Iterated logarithm approximations to the distribution of the largest prime divisor

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.