Prime Numbers

1.5 Exercises 49
Theorem 1.4.9 (Dickman). For each fixed real number u>0, there is a
real number ρ(u) > 0 such that
ψ(x, x 1/u ) ∼ ρ(u)x.
Moreover, Dickman described the function ρ(u) as the solution of a certain
differential equation: It is the unique continuous function on [0, ∞) that
satisfies (A) ρ(u) = 1 for 0 ≤ u ≤ 1 and (B) for u>1, ρ ′ (u) =−ρ(u − 1)/u.
In particular, ρ(u) =1− ln u for 1 ≤ u ≤ 2, but there is no known closed form
(using elementary functions) for ρ(u) foru>2. The function ρ(u) canbe
approximated numerically (cf. Exercise 3.5), and it becomes quickly evident
that it decays to zero rapidly. In fact, it decays somewhat faster than u −u ,
though this simple expression can stand in as a reasonable estimate for ρ(u)
in various complexity studies. Indeed, we have
ln ρ(u) ∼−u ln u. (1.43)
Theorem 1.4.9 is fine for estimating ψ(x, y) whenx, y tend to infinity with
u = lnx/ ln y fixed or bounded. But how can we estimate ψ 1/ ln ln x, x x
or ψ x, e √
ln x or ψ x, ln 2 x ? Estimates for these and similar expressions
became crucial around 1980 when subexponential factoring algorithms were
first being studied theoretically (see Chapter 6). Filling this gap, it was shown
in [Canfield et al. 1983] that
ψ x, x 1/u
= xu −u+o(u)
(1.44)
uniformly as u →∞and uln 1+ɛ x and x is large. (We have
reasonable estimates in smaller ranges for y as well, but we shall not need
them in this book.)
It is also possible to prove explicit inequalities for ψ(x, y). For example,
in [Konyagin and Pomerance 1997] it is shown that for all x ≥ 4 and
2 ≤ x 1/u ≤ x,
ψ x, x 1/u
≥ x
ln u . (1.45)
x
The implicit estimate here is reasonably good when x 1/u =ln c x,withc>1
fixed (see Exercises 1.72, 3.19, and 4.28).
As mentioned above, smooth numbers arise in various factoring algorithms,
and in this context they are discussed later in this book. The computational
problem of recognizing the smooth numbers in a given set of integers
is discussed in Chapter 3. For much more on smooth numbers, see the new
survey article [Granville 2004b].
1.5 Exercises
1.1. What is the largest integer N having the following property: All integers
in [2,...,N − 1] that have no common prime factor with N are themselves