[其他] 數論等式一題

[JohnMash (Paul)]

http://m.imgur.com/5gC8fb4

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→ kerwinhui   : sum_{p^m<x, m>1} log(p)/p^m = O(1) 可以拿走
→ kerwinhui   : summation-by-parts sum (log(p)/p)(1/log(p))
→ kerwinhui   : => 1+O(1/(log x))+log(log x)-log(log 2)+integral
→ kerwinhui   : integral=int_2^x O(1)/(t (log t)^2) = O(1/log x)
→ kerwinhui   : 得證
→ JohnMash    : 了解。謝謝。
→ JohnMash    : 要用到Shapiro's Theorem