Definition:Legendre's Constant

Definition
Legendre's constant (or the Legendre constant) is a mathematical constant conjectured by Adrien-Marie Legendre to specify the prime-counting function $\pi \left({n}\right)$.

Legendre conjectured in 1796 that $\pi \left({n}\right)$ satisfies:
 * $\displaystyle \lim_{n \to \infty} \pi \left({n}\right) - \frac n {\log(n)} = B$

where $B$ is Legendre's constant.

If such a number $B$ exists, then this implies the Prime Number Theorem.

Legendre's guess for $B$ was about $1.08366$.

Later, Gauss looked at this problem and thought that $B$ might actually be lower.

In 1896, Hadamard and Poussin independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$.

Legendre's first guess of $1.08366 \ldots$ is still (incorrectly) referred to as Legendre's constant, even though its "correct" value is in fact exactly $1$.

Hence it is only now of historical importance.