Dickman-de Bruijn function
This function, called Dickman's function or the Dickman-de Bruijn function, is defined as the function satisfying the delay differential equation:
subject to the initial condition for . The function satisfies the following properties:
- for .
- for .
- is (strictly) decreasing for , i.e., for .
- is once differentiable on . More generally, $\rho</math> is times differentiable everywhere except at the points .
- is infinitely differentiable except at integers.
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It turns out that the density of numbers with no prime divisor greater than the root is given by . Formally, consider, for any , the fraction of natural numbers such that all prime divisors of are at most . Then, as , this fraction tends to .