In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited:


[1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti dell'accademia delle scienze fisico-matematiche di Napoli vol 3 ser 8, 132-166.


My problem is that I can't find this paper.


I know that in [1] the following expansion is shown:


$\log(\log(n))=\log\_2(n)$
$$\frac{p\_n}{n}= \log(n)+\log\_2(n)-1+\sum\_{i=1}^m(-)^{i+1}\frac{P\_i(\log\_2(n))}{i! \log(n)^i}+o\Big(\frac{1}{\log(n)^m}\Big) $$ 


in publications such as [2], [3] it is written that Cipolla in [1] proved that $ P\_i (\log\_2 (n)) $ are polynomials with integer coefficients and gave a recurring formula for $ P\_i $


in [2] section 5. this recurring formula is shown:


$X=\log\_2(n)$


$$ a) \ \ P\_1(X)=X-2, \ \ P\_2(X)=X^{2}-6X+11 $$
$$ b) \ \ P'\_k=k(k-1)P\_{k-1}+kP'\_{k-1}, \ \ k\geq2$$
$$ c) \ \ P\_{k+1}(0)=-k\Biggl(\sum\_{j=1}^{k-1}\binom{n+1}{2k}P\_j(0)\Big(P\_{k-j}(0)+P'\_{k-j}(0)\Big) +P\_{k}(0)+P'\_{k}(0)\Biggl)-(k+1)P\_{k}(0)-P'\_{k+1}(0)$$


and 7 polynomials are computed,some examples:


$P\_3(X)=2X^3 -21X^2+84X-131$


$P\_4(X)=6X^4-92X^3+588X^2-1908X+2666$


$P\_5(X)=24X^5-490X^4+4380X^3-22020X^2+62860X-81534$


but $ (b), (c) $ are wrong and do not give the polynomials that are shown instead.


the error in $ (b) $ is easy to find, the correct $ b $:


$$ P'\_k=k(k-1)P\_{k-1}-kP'\_{k-1}, \ \ k\geq2$$
but I can not fix $ (c) $ which seems to be correct only up to $ P\_3 (0) = - 131 $


Can someone show me a correct version of $ c $?


Thanks for your attention


[2] <https://www.emis.de/journals/JIPAM/images/153_02_JIPAM/153_02.pdf>


[3]<https://www.researchgate.net/profile/Rafael_Jakimczuk/publication/269027863_An_observation_on_the_Cipolla%27s_expansion/links/5a2b530e0f7e9b63e538d2fe/An-observation-on-the-Cipollas-expansion.pdf>



Cipolla's Prime numbers function: Computing the coefficients of the polynomial
According to the OP, the formula requested is the formula (4.7) in the paper :


*Arias de Reyna, Juan; Toulisse, Jérémy*, [**The $n$-th prime asymptotically**](http://dx.doi.org/10.5802/jtnb.847), J. Théor. Nombres Bordx. 25, No. 3, 521-555 (2013). [ZBL1298.11093](https://zbmath.org/?q=an:1298.11093).


