In the paper <http://rgmia.org/papers/v8n2/eepnt.pdf>, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.


$$
p\_n = n \frac{\Gamma'(n)}{\Gamma(n)} + o(n \ln n).
$$


I obtained a stronger form of this result namely


$$
p\_n = n \ln \frac{\Gamma'(n)}{\Gamma(n-1)} + O\Big(\frac{n\ln\ln n}{\ln n}\Big).
$$


The gamma function seems to beautifully approximate $p\_n$. To get the same error term using the regular Cipolla's asymptotic expansion of the $p\_n$ we would need three terms.


Can someone explain why the gamma function approximated the n-th prime so nicely? Is this a coincidence or is there some underlying phenomenon governing this result that can shed some new light distribution of prime numbers. 



Asymptotics of the n-th prime using the gamma function
The asymptotic expansion of Cipolla starts
$$p\_n=n\log n+n\log\log n-n+n\frac{\log\log n}{\log n}+O(n(\log\log n/\log n)^2)$$
So the given approximations have errors 
$$p\_n=n\frac{\Gamma'(n)}{\Gamma(n)}+\Theta(n\log\log n)$$
and 
$$p\_n=n\log\frac{\Gamma'(n)}{\Gamma(n-1)}+\Theta(n).$$
I would not say these are good approximations with so big errors. 


The inverse function of the log integral function $\text{li}^{-1}(x)$ has error
$$p\_n= \text{li}^{-1}(n) +O(n \exp(-c\sqrt{\log n})$$
which assumming Riemann hypothesis can be reduced to 
$$|p\_n-\text{li}^{-1}(n)|\le \pi^{-1} \sqrt{n}(\log n)^{\frac52}\qquad n>11.$$
(see arXiv:1203.5413)


