14 1. Arithmetic Functions yields n≤x Λ(n) n = T(x) x + O ψ(x) x = log(x) + O(1) by Stirling’s formula and Proposition 1.1. The series p ∞ k=2 log(p) pk converges, and Λ(n) is zero off the prime powers. Sometimes it is necessary to remove a factor from the terms of a sum. This is an important application of the partial summation formula, and is illustrated in the proof of the next result. Proposition 1.10. The estimate p≤x 1 p = log log(x) + a + O 1 log(x) holds with some constant a. Proof. First p≤x 1 p = p≤x log(p) p 1 log(p) = ⎛ ⎝ p≤x log(p) p ⎞ ⎠ 1 log(x) − x 2 ⎛ ⎝ p≤u log(p) p ⎞ ⎠ (−1) du u log2(u) by partial summation. Then p≤x 1 p = 1 + O 1 log(x) + x 2 du u log(u) + x 2 ⎛ ⎝ p≤u log(p) p − log(u)⎠ ⎞ du u log2(u) = 1 + O 1 log(x) + log log(x) − log log(2) + ∞ 2 ⎛ ⎝ p≤u log(p) p − log(u)⎠ ⎞ du u log2(u) + ∞ x O(1) u log2(u) du = log log(x) + a + O 1 log(x) by Proposition 1.9 and integration by parts.
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