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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

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  • $\begingroup$ Did you try to write to the author of [2]? $\endgroup$
    – Francesco Polizzi
    Commented Feb 25, 2019 at 14:37
  • $\begingroup$ Anyway, you should be able to obtain a copy of Cipolla's paper via "prestito interbibliotecario" (inter-librarian loan) through the library of your University. Here is a list of italian libraries having the journal you are looking for. worldcat.org/title/… $\endgroup$
    – Francesco Polizzi
    Commented Feb 25, 2019 at 14:44
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    $\begingroup$ Just in case you do not know it already, the following paper (in particular the discussion at the beginning) seems relevant numdam.org/article/JTNB_2013__25_3_521_0.pdf $\endgroup$
    – Thomas Sauvaget
    Commented Feb 25, 2019 at 15:13
  • $\begingroup$ @ThomasSauvaget the formula (4.7) of the article you shared is the answer to my question. thank you $\endgroup$
    – Patrick Danzi
    Commented Feb 25, 2019 at 15:57
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    $\begingroup$ @ThomasSauvaget sure $\endgroup$
    – Patrick Danzi
    Commented Feb 25, 2019 at 16:02

1 Answer 1

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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, J. Théor. Nombres Bordx. 25, No. 3, 521-555 (2013). ZBL1298.11093.

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