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Let $p_n$ be the $n$-th prime. Wikipedia gives the following known bounds on $p_n/n$ when $n\geq6$: $$ \log n+\log\log n-1 \leq \frac{p_n}{n} \leq \log n+\log\log n. $$

If I take the first few terms in the asymptotic expansion for $p_n/n$, like so: $$ \frac{p_n}{n} = \log n+\log\log n-1+\frac{\log\log n}{\log n} - \frac{2}{\log n} + O\left(\frac{(\log\log n)^2}{\log^2 n}\right), $$ it follows that $$ \frac{p_n}{n} < \log n+\log\log n-1+\frac{\log\log n}{\log n} + \frac{c}{\log n}, $$ for $c>-2$ and large enough $n$. For at least those $n$ that I've checked however ($p_n\leq 10^{11}$), I find that this inequality holds with $c=0$ and $p_n>347$, and also with $c=-1$ and $p_n > 5393$.

Is this actually correct?

Is there a sharper inequality?

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    $\begingroup$ As a general warning, $10^{10}$ is a relatively small number in regards to conjectures like these. $\endgroup$
    – user14972
    Jul 2 '13 at 5:19
  • $\begingroup$ Fair enough, I just ran out of memory. $\endgroup$
    – Kirill
    Jul 2 '13 at 5:25
  • $\begingroup$ A recent paper : de Reyna and Jeremy 'The n-th prime asymptotically'. $\endgroup$ Jul 2 '13 at 23:37
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    $\begingroup$ @RaymondManzoni Thank you. I followed their reference to arxiv.org/abs/1002.0442, a paper by Pierre Dusart, which has the desired inequality on page 2. Can you submit your comment as an answer, mentioning the Dusart paper, so that I can accept it? $\endgroup$
    – Kirill
    Jul 3 '13 at 0:21
  • $\begingroup$ I am Glad the link was useful @Kirill and provided an answer (I didn't know then that Dusart's fine work was exactly the answer you wanted since it was 2am here...). $\endgroup$ Jul 3 '13 at 10:05
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While examining de Reyna's work about Riemann's zeros I noticed his recent paper with Jeremy 'The n-th prime asymptotically' and thought it could help you (without having the time to elaborate...).

Knowing more zeta zeros allowed indeed Pierre Dusart in his $1998$ thesis to update Rosser and Schoenfeld's earlier results and get new bounds on $\pi(x)$ (other results are here in french).

He extended his results to bounds for $p_n$ in his $1999$ paper 'The Kth prime is greater than k(ln(k)+ln(ln(k))-1) for k>=2' and in $2010$ in the paper you found 'Estimates of Some Functions Over Primes without R.H.' and which contains indeed the precise : $$\frac{p_n}n\le\ln(n)+\ln(\ln(n))-1+\frac{\ln(\ln(n))-2}{\ln(n)}\quad\text{for}\ n\ge 688383,$$ $$\frac{p_n}n\ge\ln(n)+\ln(\ln(n))-1+\frac{\ln(\ln(n))-21/10}{\ln(n)}\quad\text{for}\ n\ge 3.$$

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