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Exercise 2.1.4. Complete the proof of Theorem 2.1.3(c) as follows: (a) For  ∈ (0, 1] and x ≥ 2, show that X p≤x log  1 − 1 p  = X p≤x log  1 − 1 p 1+/ log x  + O(). (b) For s ∈ (1, 2], show that X p log  1 − 1 p s  = log(s − 1) + O(|s − 1|). (c) For  ∈ (0, 1] and x ≥ 2, show that X p>x log  1 − 1 p 1+/ log x  = − Z ∞  e −u u du + O  1 log x  . [Hint: Theorem 2.1.3(a).] Conclude that X p≤x log  1 − 1 p  = − log log x + Z ∞ 0 e −u − 1[0,1](u) u du + O   + 1 log x  , where 1A denotes the characteristic function of the set A. (d) Note that γ = lim N→∞  − log N + Z 1 0 (1 + x + · · · + x N−1 )dx . Make the change of variables x = 1 − u/N and deduce that γ = R ∞ 0 1[0,1](u)−e−u u du to complete the proof of Theorem 2.1.3(c). Exercise 2.2.5. Given x, y ≥ 1, we define Φ(x, y) = #{n ≤ x : p|n ⇒ p > y}. Prove that there is a constant c ≥ 2 such that Φ(x, y)  x log y (2 ≤ y ≤ x/c) Exercise 2.3.5. (a) Show that X n≤x log(x/n)  x (x ≥ 2). (b) Prove that µ log = −µ ∗ Λ and deduce that M(x) log x = − X d≤x Λ(d)M(x/d) + O(x). (c) Show that lim inf x→∞ M(x) x + lim sup x→∞ M(x) x = 0.

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