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Dickman's function
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Definition/Summary
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| Dickman's function, or the Dickman-de Bruijn function, is an estimate to the fraction [itex]\alpha[/itex] of the number of [itex]x^\alpha[/itex]-smooth numbers below x. |
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Equations
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| [tex]u\rho'(u) + \rho(u-1) = 0[/tex] |
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Scientists
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| Dickman and Ramaswami |
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Recent forum threads on Dickman's function
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Breakdown
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Mathematics
> Number Theory
>> Number Theoretic Functions
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Extended explanation
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The Dickman-de Bruijn function [itex]\rho(u)[/itex] is defined by the delay differential equation above, with the initial condition [itex]\rho(u) = 1[/itex] for [itex]0\le u\le1 [/itex].
Roughly, [itex]\rho(u)\approx u^{-u}[/itex] (also [itex]\rho(x)\le1/x![/itex]). A better estimate is
[tex]\rho(u)\sim\frac{1}{\xi\sqrt{2\pi x}}\cdot\exp(-x\xi+\operatorname{Ei}(\xi))[/tex]
where Ei is the exponential integral and ξ is the positive root of
[tex]e^\xi-1=x\xi[/tex]. |
Commentary
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