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.
