Legendre's Constant In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here).
Actually the first uses $\lim_{n \to \infty} (\log(n) - (n/\pi(n)))$, but I have the same question in either case.
To be honest, I just assumed that this was a typo for $\lim_{n \to \infty} (\pi(n) / (n/\log(n)))$ (that is, the prime number theorem). However, in both pages it looks to me as if the claim really is that $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ exists and is $1$. Really?
I don't know whether to be more surprised that the limit exists or that its value is 1
Does anyone else find it surprising?
 A: I'll stay clear from the infamous $B_L'$ notation.
Legendre's original (correct) statement is that
$$\pi(x)=\frac{Bx}{\log x-A+o(1)}$$
where the so-called "Legendre's  constant" is $A$. The incorrect part was that he guessed that $A=1.08366$
As pointed out by Fedor Petrov, the definition with $\pi(n) - (n/\log(n))$ is clearly wrong.

Note. Using de la Vallée Poussin to disprove Legendre's conjecture is actually an overkill. This elementary argument is due to Pintz.


*

*János Pintz, "On Legendre's prime number formula" (1980)


Let
$$\Psi(x)=Cx+\frac{(D+o(1))x}{\log x}$$
with some constants $C$ and $D$. Using Stirling and Legendre's formula,
\begin{equation*}
\begin{split}
\log x+O(1) &= \frac{\log [x]!}{x}=\frac{1}{x} \sum_{n\leq x} \Lambda (n) \left[ \frac{x}{n} \right]\\ 
& = \sum_{n\leq x} \frac{\Lambda (n)}{n}+\frac{1}{x} \sum_{n\leq x} \Lambda (n)O(1)\\
& = \int_2^x \frac{\Psi(t)}{t^2}dt+\frac{\Psi(x)}{x}+O\left( \frac{\Psi(x)}{x}\right)\\
& = \int_2^x \frac{C+\frac{D+o(1)}{\log t}}{t}dt+O(1)\\
& = C\log x + (D+o(1))\log\log x.
\end{split}
\end{equation*}
Therefore $C=1$ and $D=0$, that is,
$$\Psi(x)=x\left(1+\frac{o(1)}{\log x}\right).$$
And by partial summation we finally get $A=B=1$ in Legendre's formula.
A: 1. The prime number theorem in the form
$$\pi(n)=\mathrm{li}(n)+O(ne^{-c\sqrt{\log n}})$$
combined with the approximation
$$\mathrm{li}(n)=\frac{n}{\log n}+\frac{n}{\log^2 n}+O\left(\frac{n}{\log^3 n}\right)$$
shows that
$$\pi(n)-\frac{n}{\log n}=\frac{n}{\log^2 n}+O\left(\frac{n}{\log^3 n}\right).$$
So the left hand side tends to infinity quite rapidly, it has no finite limit. 
2. The correct definition of Legendre's constant is
$$A:=\lim_{n\to\infty}\left(\log n-\frac{n}{\pi(n)}\right),$$
and the third display above shows that it equals $1$:
\begin{align*}\log n-\frac{n}{\pi(n)}
&=\log n-\frac{n}{\frac{n}{\log n}+\frac{n+o(n)}{\log^2 n}}\\
&=\log n-\frac{\log n}{1+\frac{1+o(1)}{\log n}}\\
&=(\log n)\left(1-\frac{1}{1+\frac{1+o(1)}{\log n}}\right)\\
&=(\log n)\left(1-1+\frac{1+o(1)}{\log n}\right)\\
&=(\log n)\frac{1+o(1)}{\log n}\\
&=1+o(1).\end{align*}
