Problems on Chapter 1 17 (a) a≤x a∈A L(a) = (1 + o(1))x as x → ∞, (b) A(x) = (1 + o(1)) x a1 dt L(t) as x → ∞. Problem 1.12. Let L : [M, +∞) → R+, where M 0. Assume that L ∈ C[M, +∞). Prove that L ∈ L ⇐⇒ x M dt L(t) = (1 + o(1)) x L(x) as x → ∞. Problem 1.13. Let A ⊂ N, with a1 being its smallest element. Let L : [a1, +∞) → R+ be an increasing slowly oscillating function. Prove that lim x→∞ 1 x a≤x a∈A L(a) = 1 ⇐⇒ lim x→∞ L(x) x a≤x a∈A 1 = 1. Problem 1.14. Let A = {p : p + 2 is prime}. It is conjectured that A(x) ∼ Cx/ log2 x (as x → ∞) for a certain positive constant C. Use the preceding problem to show that this conjecture implies that p≤x p+2 prime 1 C log2 p = (1 + o(1))x (x → ∞). Problem 1.15. Prove that 1 ≤ n! nne−n √ 2πn ≤ e1/12n (n ≥ 1).

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