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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