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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.