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.
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