40 1. Arithmetic Functions

(56) † Let c ≥ 1 be a fixed real number, and let fc(n) be the number of

divisors d of n satisfying the inequality 1/c ≤

n/d2

≤ c. Show that

n≤x

fc(n) = x log(c) + O(

√

cx)

where the constants implied in the O-term do not depend on c. (The

estimate is uniform in c.) Then conclude that 50% of the divisors d of

the integers n in the interval 1 ≤ n ≤ x satisfy the inequality

1

√

x

≤

n

d2

≤

√

x

as x → +∞. An estimate that contains a parameter frequently becomes

much more useful if we can establish that the estimate is uniform in the

parameter.