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I saw this formula $$\frac{p_n}{n} < \log n + \log \log n \quad\text{for } n \ge 6$$ on this Wikipedia site https://en.wikipedia.org/wiki/Prime_number_theorem#Non-asymptotic_bounds_on_the_prime-counting_function. I needed a proof so I checked the sources. I couldn't find a proof...

If there is, please tell me on what page and I apologize deeply. If not - please send me a link to a proof.

J. W. Tanner
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Daniel
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  • No, I have already seen this post. The proof should be here: https://projecteuclid.org/download/pdf_1/euclid.ijm/1255631807 (page 69), but he only states it without a proof. – Daniel Aug 03 '20 at 22:11
  • You are right, I am confused that this paper is not in the linked answer so I have added an answer myself. Anyway, on page 83 in the paper you linked there is a proof of your desired result – Maximilian Janisch Aug 03 '20 at 22:22
  • Yes but how can we determine how small or large $n$ can be? – Daniel Aug 03 '20 at 22:28
  • I haven't looked at the intricacies of the proof. But it seems to me like they are using a combination of tables and number theoretic arguments to arrive at $$\frac{p_n}n<\ln(n)+\ln(\ln(n))-\frac12$$ for $n\geq20$ from which your result readily follows. – Maximilian Janisch Aug 03 '20 at 22:31
  • Okay thank you. One last question: Do you know a boundary, which is even better? – Daniel Aug 03 '20 at 22:35
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    Sorry, I know nothing better from the top of my head. But I am sure you can find even sharper bounds in the literature. – Maximilian Janisch Aug 03 '20 at 22:37

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