Approximation of the n'th prime

Question

An old paper by Ernest Cesàro provides a suggested approximation of the n'th prime. The expression and the reference currently appears in the Wikipedia article on the Prime Number Theorem.

It is inspired by an even earlier and difficult to locate article by one Monsieur Pervouchine aka Ivan Mikheevich Pervushin in a Russian Journal.

Cesàro points out that his own original approximation that appeared in a paper in Actes de l'Académie des Sciences de Naples in 1893: $$\frac{p_n}{n} = \log p_n - 1 - \frac{1}{\log p_n} - \frac{3}{(\log p_n)^2} - \cdots$$ (presumably with the implicit understanding that all subsequent terms add up to $o(1/(\log p_n)^2)$) implies Pervushin's approximation.

What I would like to know is whether there exists a modern or at least better accessible account of this type of approximation. And I am curious why it seems much easier to get the attempted asymptotics right numerically when I change the "3" to a "2"?

Answer 1

A more modern account (from 1962, though still widely cited) is Rosser and Schoenfeld's classic paper Approximate formulas for some functions of prime numbers. See Theorem 3 for example.

Answer 2

In this paper, it is shown that the prime counting function $\pi(x)$ satisfies $$ \pi(x)=\frac{x}{\log x-1-\frac{k_1}{\log x} - \frac{k_2}{\log^2 x} -\ldots-\frac{k_m}{\log^m x}+o\!\left(\frac{1}{\log^m x}\right) } $$ for any fixed $m\ge 1$ as $x\to+\infty$. Here $$ k_m+1!\cdot k_{m-1}+2!\cdot k_{m-2}+\ldots+(m-1)!\cdot k_1=m\cdot m! $$ for $m\ge 1$. In particular, $k_1=1$, $k_2=3$, $k_3=13$, $k_4=71$. Substituting $x=p_n$ and re-arranging the resulting equality yields $$ \frac{p_n}{n}=\log p_n-1-\frac{k_1}{\log p_n} - \frac{k_2}{\log^2 p_n} -\ldots-\frac{k_m}{\log^m p_n}+o\!\left(\frac{1}{\log^m p_n}\right) $$ for any fixed $m\ge 1$ as $n\to+\infty$. This is a rigorous interpretation of Cesàro's result.

Addendum. An explicit asymptotic expansion comes from the asymptotic inversion of the (offset) logarithmic integral $\operatorname{Li}(x)$: $$ \frac{p_n}{n} \sim \log n + \sum\limits_{k = 0}^\infty {\frac{{P_k (\log \log n)}}{{\log ^k n}}} $$ as $n\to+\infty$, where $P_0 (x) = x - 1$, $P_1 (x) = x - 2$, and \begin{align*} P_k (x) = kP_{k - 1} (x) & - P'_{k - 1} (x)\\ & + \frac{1}{k}\sum\limits_{m = 1}^{k - 1} {m((m - 1)P_{m - 1} (x) - P_m (x) - P'_{m - 1} (x))P_{k - m - 1} (x)} \end{align*} for $k\ge 2$. This result is originally due to Cipolla. The recurrence for the polynomials $P_k(x)$ was derived in this paper.