Is this published upper bound on the n'th prime wrong?

Question

This paper gives an upper bound on the n'th prime for $n\ge7022$ as:$$p_n\le n(\log{n} + \log{\log{n}}-0.9385)$$ citing the reference Robin, Guy, Estimate of the Chebyshev theta function on the $k$th prime number and large values of the number of prime divisors function $\omega(n)$ of $n$, Acta Arith. 42, 367-389 (1983). ZBL0475.10034. This paper in turn attributes the expression to Rosser in this paper (one that I have not been able to access).

But $p[8597]=88789$ whereas the upper bound given by the above expression is $\approx88759$. I am wondering if the original paper contains an error or whether the original expression has been wrongly quoted. Could anyone please enlighten me?

The article is available here: https://sci-hub.mn/https://doi.org/10.1112/plms/s2-45.1.21# — Enrico M., Dec 16 '17 at 13:06

Dietrich Burde · Accepted Answer · 2017-12-16T13:37:51.670

4

The upper bounds for $p_n$, and estimates of $\pi(x)$ are often incorrect, see also the remarks in this article. I can confirm your computation. I think, Dusart's estimates are more reliable, see here. It seems to me that Rosser's paper is correct.

edited Dec 16 '17 at 13:37

answered Dec 16 '17 at 13:09

Dietrich Burde

130,978

Thank you for your input and for your informative article. – BlindAnagram Dec 16 '17 at 14:47

Is this published upper bound on the n'th prime wrong?

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