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Then any integer n can be written uniquely as n = nP m where nP ∈ P and m is coprime to p for all p ∈ P. Show that d dn d∈ P equals 1 if nP = 1 and zero otherwise. If P n is defined to be log p whenever n = pa for some prime p ∈ P and zero otherwise, then show that P n =− d log d dn d∈ P whenever nP > 1. 20. With notation as in the previous exercise, let A be a set of natural numbers ≤ x and let S A P denote the set of elements n of A with nP = 1. 5 Exercises for some 31 and some R x d with R x d log d≤x d∈ P x =O x d (ii) if n ∈ A with nP > 1, then nP has at least two prime factors counted with multiplicity; (iii) there is a set B such that S A P = S B P and satisfying the condition that for p ∈ P and m ∈ B, we have pm ∈ B; (iv) there are numbers a and b with a > 0 so that 1 1 = a log x + b + O m x m≤x m∈B Under these conditions, show that for some positive constant c, cx #S A P ∼ log x 1−a/b as x tends to infinity.

16. ] u = 1 − log u + log v − 1 Some elementary sieves 30 17. Define u recursively by integers k, by u = u = 1 for 0 ≤ u ≤ 1 and, for positive k − u k v−1 dv v for k < u ≤ k + 1. Using Buchstab’s identity, deduce inductively that for any > 0 and x < y ≤ x, we have the asymptotic formula x y ∼x u with u = log x/ log y. [ u is called Dickman’s function and was discovered by K. Dickman in 1930. For further details on this function, as well as more results concerning x y , we refer the reader to [68, p.

5 Exercises 1. Prove that n − log log n 2 = O x log log x n≤x 2. Let y n denote the number of prime divisors of n that are less than or equal to y. Show that y n − log log y 2 = O x log log y n≤x 3. Prove that n − log log x 2 = x log log x + O x n≤x 4. Let that n denote the number of prime powers that divide n. Show n has normal order log log n. The normal order method 44 5. Fix k ∈ and let a k = 1. Denote by n k a the number of prime divisors of n that are ≡ a mod k . Show that n k a has normal order 1 log log n k 6.