# Cipolla's Prime numbers function: Computing the coefficients of the polynomial

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

: 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  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 ,  it is written that Cipolla in  proved that $$P_i (\log_2 (n))$$ are polynomials with integer coefficients and gave a recurring formula for $$P_i$$

in  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 $$c$$?

Arias de Reyna, Juan; Toulisse, Jérémy, The $$n$$-th prime asymptotically, J. Théor. Nombres Bordx. 25, No. 3, 521-555 (2013). ZBL1298.11093.