Using familiar lemma

Calculus Level 4

Let: \(\displaystyle L=\large \lim_{n\to\infty} \sum_{k=1}^n \dfrac{1}{n+\pi(k)\ln n}\)

where \(\pi(k)\) denotes the number of prime not exceeding \(k\).

Find \(\left\lceil 1000L\right\rceil\).

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