Iterated logarithm approximations to the distribution of the largest prime divisor

Details Iterated logarithm approximations to the distribution of the largest prime divisor Iterated logarithm approximations to the distribution of the largest prime divisor

2009-03-16

arxiv.org/abs/0903.2712v1

Mathematics

Number Theory

44 pages

Scientific paper

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}

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