Using familiar lemma

Calculus Level 4

Let \(\displaystyle L= \lim_{n\to\infty} \sum_{k=1}^n \dfrac{1}{n+\pi(k)\ln n},\) where \(\pi(k)\) denotes the number of primes not exceeding \(k.\)

Find the closed form of \(L \) and submit your answer as \(\big\lceil 1000L\big\rceil.\)

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