For those without access to Silverman and Wagstaff's paper "A Practical Analysis of Elliptical Curve Factoring,"

*Mathematics of Computation*, Vol. 61, No. 203, (July 1993), p 445-462, it defines rho(alpha) and mu(alpha, beta) as

rho(alpha) = probability largest factor of x is less than x

^{(1/alpha)}
mu(rho, alpha) = probability second largest factor of x is less than x

^{(1/alpha)}
AND

largest factor is less than x

^{(beta/alpha)}.

Rho is a commonly used version of the Dickman function, with many internet references, including

Mathworld. In particular, it's known that

for 0 <= t <= 1 rho(t) = 1

for 1 <= t <= 2 rho(t) = 1-ln(t)

The pdf version of the paper (I haven't seen a print version, so this may a conversion error) then claims that mu is defined by the formula below. As explained in this thread, the forumla is wrong and does not match the tables lin the paper.