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For an asymptotic estimate for the count of $y$-smooth numbers between $0$ and $x$, with the Dickman-de Bruijn function $\Psi(x, y) \approx x \rho(u)$ where $u = log( x)/log( y)$, Hildebrand and Tenenbaum provides the simple bounds for $\rho(u)$:

$0 < \rho(u) < 1/ \Gamma(u + 1)$.

Does an upper bound for $\rho(u)$ necessarily imply that $x/\Gamma(a+1)$ is an upper bound for $\Psi(x, y)$? My preliminary data suggests that this is true. If so, is it because $x\rho(u)$ is asymptotic that makes this true?

If not, I think I have some brute force checks to do to compare this function versus known upper bound formulas.


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An upper bound for $\rho(u)$ should not necessarily imply an upper bound for $\Psi(x,y)$. Pomerance asked whether it is true that $\Psi(x,y)\geq x\rho(u)$ for all $x\geq 2y\geq 2$. Thus a non-trivial lower bound for $\rho(u)$ might imply a non-trivial lower bound for $\Psi(x,y)$. The upper bound for $\rho(u)$ is rather weak, and thus in a rather large $xy$-region your upper bound for $\Psi(x,y)$ might be true, but this requires proof...

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Pieter: I've gone ahead and TeXified your answer; have a look and let me know if I've made any errors. In general, TeX in answers works much like math-mode TeX everwhere else, including the \$ and \$\$ delimiters; if you have any other math-mode questions, check out the guideline at… . – Steven Stadnicki Oct 30 '12 at 17:31

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